SCALAR CURVATURE, METRIC DEGENERATIONS AND THE STATIC VACUUM EINSTEIN EQUATIONS ON 3-MANIFOLDS, I.

MICHAEL T. ANDERSON

Contents

0. Introduction. 1 1. Background Material. 6 2. Initial Global Estimates for Yamabe Metrics. 17 3. Existence of Non-Flat Blow-Ups. 26 4. Remarks on the Hypotheses. 53 5. Completeness of the Blow-up Limits. 58 6. Construction of Yamabe Sequences with Singular Limits. 65 7. Palais-Smale Sequences for Scalar Curvature Functionals. 84 8. Appendix. 88 References 93

0. Introduction

In this paper, we prove that degenerations of sequences of Yamabe metrics on 3-manifolds are modeled or described by solutions to the static vacuum Einstein equations. One underlying motivation to understand such degenerations is the question of existence of constant curvature metrics on 3-manifolds, in other words with the geometrization conjecture of Thurston [Th2]. An approach towards resolving this conjecture via study of Yamabe metrics is outlined in [An1].

Let 𝕄 denote the space of all smooth Riemannian metrics on a closed, oriented 3-manifold M, and 𝕄_{1} the subset of metrics satisfying vol_{g} M = 1. Define the total scalar curvature or Einstein-Hilbert action S : 𝕄 → ℝ by

S(g) = v^{−1/3} ∫_{M} s_{g}dV_{g},  (0.1)

where s_{g} is the scalar curvature of the metric g, dV_{g} is the volume form associated with g and v is the volume of (M, g). The critical points of S are Einstein metrics, i.e. metrics satisfying the equation

−∇S|_{g} = v^{−1/3}z = v^{−1/3} (r − \frac{s}{3} · g) = 0,  (0.2)

Partially supported by NSF Grants DMS 9505744 and 9802722.

where r is the Ricci curvature of g and z is the traceless Ricci curvature. In dimension 3, (and only in this dimension), Einstein metrics are exactly the metrics of constant curvature.

There is a well-known minimax procedure to obtain critical values of S. First, the solution to the Yamabe problem [Y], [Tr], [Au1], [Sc1] implies that in the conformal class [g] of any metric g ∈ 𝕄_{1}, there is a metric ḡ ∈ 𝕄_{1} which realizes the infimum μ[g] of S|_{[g]∩𝕄_{1}}, i.e.

sḡ = μ[g] ≡ inf S|_{[g]∩𝕄_{1}}.  (0.3)

Such metrics are called Yamabe metrics. Let C denote the space of Yamabe metrics in 𝕄, (of arbitrary volume), and C_{1} the subset of unit volume Yamabe metrics; any metric in C can be scaled uniquely to a metric in C_{1}. In dimension 2, the space C_{1}, when divided by the action of the diffeomorphism group, corresponds to the Teichmüller or moduli space of a surface, and the functional sḡ : C_{1} → ℝ is a constant function, (by the Gauss-Bonnet theorem). In dimension 3, (and above), this is an infinite dimensional space, and the functional is not constant. A simple but important comparison argument of Aubin [Au1] implies that, for any conformal class [g] on any M,

μ[g] ≤ μ(S^{3}, g_{can}),  (0.4)

where g_{can} is the canonical constant curvature metric on S^{3} of volume 1. Define the Sigma constant σ(M) of M by

σ(M) = sup_{C_{1}} μ[g].  (0.5)

This is a smooth invariant of the 3-manifold M, which should act very much like an Euler characteristic for 3-manifolds, when appropriately normalized. In case σ(M) ≤ 0, it is easy to prove, c.f. [Bes, Prop.4.47], that any Yamabe metric g_{o} ∈ C_{1} such that s_{g_{o}} = σ(M) is Einstein. In case σ(M) > 0, this has been conjectured to be true, c.f. [Bes, Remark4.48], but remains still unknown.

Of course, an arbitrary closed, oriented 3-manifold does not admit an Einstein metric; this is the case if for instance M has an essential 2-sphere. Thus, if M is an arbitrary closed 3-manifold and if {g_{i}} is a maximizing sequence of unit volume Yamabe metrics on M, that is

s_{g_{i}} → σ(M),  (0.6)

then in general this sequence must degenerate in some manner.

More specifically, if {g_{i}} satisfies the uniform curvature bound

∫_{M} |z_{g_{i}}|^{2}dV_{g_{i}} ≤ Λ,  (0.7)

for some Λ < ∞, then as outlined in [An1], the structure of a suitable subsequence of {g_{i}} and its limit can be used to geometrize the 3-manifold M, (at least when σ(M) ≤ 0). This is accomplished essentially via the Cheeger-Gromov theory of convergence and collapse of Riemannian manifolds [C], [G, Ch.8], [CG1,2]. Since an arbitrary 3-manifold cannot be given a geometric structure, a maximizing sequence {g_{i}} of unit volume Yamabe metrics must in general satisfy

∫_{M} |z_{g_{i}}|^{2}dV_{g_{i}} → ∞, as i → ∞.  (0.8)

For the purposes of this paper, we will say that a sequence {g_{i}} satisfying (0.8) degenerates. One then would like to relate the geometry of a degenerating sequence of Yamabe metrics with some topological structure on the underlying manifold M that is the cause for the degeneration, c.f. [An1, §4-6] for further discussion.

The main purpose of this paper is to show that the degenerations of such a sequence {g_{i}} are described by solutions to the static vacuum Einstein equations. Further, under natural conditions, the degenerations correspond to non-trivial solutions of these equations.

Some further background is needed to explain this; we refer to §1-§2 for further details. Let L be the linearization of the scalar curvature function at g given by

L(α) = \frac{d}{dt}s(g + tα)|_{t=0} = −Δ tr α + δδα − ⟨r, α⟩,  (0.9)

c.f. [Bes, Ch.1K]. The L^{2} adjoint L^{∗} of L is given by

L^{∗}(f) = D^{2}f − Δf · g − f · r.  (0.10)

For any g ∈ C, one thus has a natural splitting

T_{g}𝕄 = T_{g}C ⊕ N_{g}C,  (0.11)

where

T_{g}C = {α ∈ T_{g}𝕄 : L(α) = const}, N_{g}C = {β ∈ T_{g}𝕄 : β = L^{∗}(q), ∫ q = 0} ,

are the (formal) tangent and normal spaces to C in 𝕄.

In particular, the L^{2} gradient ∇S = −v^{−1/3}z of the scalar curvature functional S splits at g as

z = z^{T} + z^{N},  (0.12)

where −v^{−1/3}z^{T} is the gradient of the scale invariant functional v^{2/3} · s = S|_{C} on C.

Another natural splitting of T 𝕄, differing from (0.10) by just a 1-dimensional factor, is

T_{g}𝕄 = Ker L ⊕ Im L^{∗}.  (0.13)

As in (0.12), we have then an L^{2} orthogonal sum

z = ξ + L^{∗}f,  (0.14)

for some ξ ∈ Ker L and f ∈ C^{∞}(M). We set

u = 1 + f.  (0.15)

Next, we describe briefly the static vacuum Einstein equations. These are equations

hr = D^{2}h,  (0.16)
Δh = 0,

for a pair (g, h) consisting of metric g and positive harmonic potential function h, defined on an open 3-manifold N. These equations have been extensively studied in classical general relativity. They imply that the 4-manifold X = N ×_{h} S^{1}, with warped product metric g_{X} = g_{N} + h^{2}dθ^{2} is Ricci-flat, i.e. a vacuum solution to the Einstein equations in dimension 4, c.f. §1.3.

Of course, a flat metric, with h an affine function, is a solution of (0.16). It is proved in the Appendix, c.f. also Theorems 3.2 and 3.3, that the only complete solution to the static vacuum equations (0.16) with potential h > 0 everywhere is a flat metric, with h = const. The most important or “canonical” (non-trivial) solution to the static vacuum equations is the Schwarzschild metric g_{s},

g_{s} = (1 − 2mt^{−1})^{−1}dt^{2} + t^{2}ds_{S^{2}}^{2},  (0.17)

defined on [2m, ∞) × S^{2}, with h = (1 − 2mt^{−1})^{1/2}; here the mass m is a positive constant. This metric is asymptotically flat, (for large t), and the locus Σ = {h = 0} = {t = 1} is a round totally geodesic 2-sphere, of radius 2m. Physically, this represents the surface of an (idealized) static black hole.

Finally, given a sequence {g_{i}} ∈ 𝕄_{1}, a sequence r_{i} ∈ ℝ converging to 0, and points x_{i} ∈ M, the blow-up sequence is the pointed sequence of Riemannian manifolds (M, ḡ_{i}, x_{i}) with ḡ_{i} = r_{i}^{−2} · g_{i}. For instance, in case the curvature tensor R_{i} of (M, g_{i}) is everywhere bounded by r_{i}^{−2}, the Cheeger-Gromov theory implies that a subsequence of the pointed sequence (M, ḡ_{i}, x_{i}) either converges, modulo diffeomorphisms, uniformly on compact subsets, to a limit complete Riemannian manifold (N, ḡ, x), or collapses along a sequence of F -structures to a lower dimensional space.

A simplified version of the main result of this paper is the following:

Theorem A. Let {g_{i}} be a sequence of unit volume Yamabe metrics on a closed oriented 3-manifold M, with s_{g_{i}} ≥ −s_{o} ≥ −∞, satisfying the following condition: (i). (Non-Collapse). There is a constant ν_{o} > 0 such that

vol B_{x}(r) ≥ ν_{o}r^{3},  (0.18)

for any geodesic ball B_{x}(r) ⊂ (M, g_{i}), r ≤ diam(M, g_{i}).

(I). Then given any sequence of points x_{i} ∈ (M, g_{i}) for which ∫_{B_{x_{i}}(r_{o})} |z_{g_{i}}|^{2}dV_{g_{i}} → ∞, for some r_{o} > 0, there is a blow-up sequence (M, g′_{i}, x_{i}),

g′_{i} = ρ_{i}^{−2} · g_{i}, ρ_{i} → 0,

such that a subsequence converges to a locally defined solution of the static vacuum Einstein equations.

(II). In addition to (i), suppose the following conditions hold: (ii). There is a constant K < ∞, such that

∫_{M} |z_{g_{i}}^{T}|^{2}dV_{g_{i}} ≤ K.  (0.19)

(iii). For u as in (0.15), there is a constant δ_{o} > 0 such that the sequence {g_{i}} degenerates on the

domain U_{δ_{o}} = \{x ∈ (M, g_{i}) : \frac{|u_{i}(x)|}{sup |u_{i}|} ≥ δ_{o}\} , i.e.

∫_{U_{δo}} |z_{g_{i}}|^{2}dV_{g_{i}} → ∞.  (0.20)

Then there are points y_{i} ∈ U_{δ_{o}} with |z_{g_{i}}|(y_{i}) → ∞, and a blow-up sequence (M, g′_{i}, y_{i}) defined as above, such that a subsequence converges to a non-trivial, (in particular non-flat), locally defined solution to the static vacuum Einstein equations.

We first make some comments on the hypotheses and conclusions. Condition (i) is used to rule out the possibility of collapse of the blow-up sequence. This condition will be weakened in §3, but some version of it is essential for Theorem A. Without any lower bound on the volumes of (small) geodesic balls, one cannot expect blow-ups to converge at all, see §4.1 for further discussion. Theorem A(I) gives then a general relationship between degenerations of Yamabe metrics and solutions, possibly trivial, of the static vacuum equations.

Conditions (ii) and (iii) are required to obtain a non-trivial limit solution. For instance, in case σ(M) < 0, condition (ii) prevents the function u, which basically serves as the potential function h in the static vacuum solution (0.16), from going to 0 in L^{2}. The condition (ii) is not as strong an assumption as it might at first appear. For example, it is proved in Theorem 2.10 that if {g_{i}} is an arbitrary sequence of unit volume Yamabe metrics on M for which there exist points x_{i} ∈ M and arbitrary numbers r_{o} ≥ 0, R_{o} < ∞, such that

∫_{B_{x_{i}}(r_{o})} |r_{g_{i}}|^{2} ≤ R_{o}, vol B_{x_{i}}(r_{o}) ≥ R_{o}^{−1},  (0.21)

then (0.19) holds, with K depending only on r_{o}, R_{o}.

Note that (0.21) requires only that {g_{i}} have locally bounded (L^{2,2}) geometry in some ball of small but fixed size in M, and yet it implies the global conclusion (0.19). This novel feature of some of the global aspects of Yamabe metrics plays a central role in this paper.

Finally, the condition (iii) that {g_{i}} degenerates on U_{δ_{o}}, for some δ_{o} > 0, and not just somewhere on (M, g_{i}) as in (0.8), is also essential to obtain a non-trivial blow-up limit solution. Examples discussed in §6 will illustrate the necessity of this condition.

A much more precise formulation of Theorem A will be proved in Theorem 3.10 below, where the base points y_{i} and scale factors ρ_{i} are constructed from the geometry of (M, g_{i}). Further, it is shown in §5 that the limit solutions are complete in a natural sense and some initial results on their asymptotic geometry are obtained.

Theorem A implies that blow-up limits of degenerations in U_{δ_{o}} of sequences of Yamabe metrics satisfy a strong, in particular a determined, set of equations. This is quite surprising, since Yamabe metrics themselves satisfy no such rigid equations; the equation defining a Yamabe metric is highly underdetermined. Note also that Theorem A holds for quite general sequences of Yamabe metrics; for example no assumption is made that the sequence {g_{i}} is a maximizing sequence for S|_{C}.

It has long been known that static vacuum solutions on an open 3-manifold N are closely tied with 2-spheres in N, occurring at the event horizon or boundary Σ of a black hole in general relativity. This is already seen in the Schwarzschild metric (0.17), and is apparent in the work of Hawking; c.f. [HE, Ch.9.3] and §1.3 for further discussion. More generally, many solutions of the static vacuum equations are asymptotically flat, and thus have natural 2-spheres near infinity. These remarks illustrate a basic tie relating the degeneration of Yamabe metrics with the underlying topology of the manifold mentioned above.

On the other hand, the first two examples discussed in §6 show that the restriction to U_{δ_{o}} in Theorem A is essential. If {g_{i}} degenerates only on the complement of U_{δ_{o}}, for any given δ_{o} > 0, then the blow-ups describing the degeneration may satisfy the static vacuum equations only trivially; see § 3.2. We also present examples in §6 of degenerating sequences which do satisfy the hypotheses of Theorem A(II), showing that this result is indeed applicable.

It is certain global features of Yamabe metrics which lead to the necessity of condition (iii) in Theorem A. In §7, we consider the relation of this condition with the condition that a sequence of Yamabe metrics is Palais-Smale for the functional S|_{C}. (Recall that a sequence is Palais-Smale for a functional if the gradient tends to 0, in a suitable (weak) topology; this is not to be confused with Condition C of Palais-Smale, which is much too strong to be of use here). In §4.2, it is shown that condition (iii) follows for sequences of Yamabe metrics satisfying a natural strengthened version of the definition of Palais-Smale sequence for S|_{C}, at least when σ(M) < 0.

In any case, in order to effectively describe the structure of degenerations of sequences of Yamabe metrics on all of M, (not only on U_{δ_{o}}), one needs to restrict to special classes of sequences. In addition to the reason above, this is also needed in case the sequence collapses, as discussed in §4.1. Some such preferred sequences are considered already in [An2, §8], and we will discuss these and related special sequences in more detail in [AnII].

This paper is partly intended as the third in a series of works on the geometrization conjecture of Thurston, the previous works being [An1, 2].

Acknowledgments: I would like to thank Dennis Sullivan and Mike Freedman, who organized and led seminars in Stony Brook and La Jolla on some of the topics treated here and the sequel papers. Thanks also to Richard Hamilton, Steve Kerckhoff, Yair Minsky, Richard Schoen, Jon Wolfson, and other participants in these seminars for their comments, and to Jeff Cheeger for his abiding support.

Many thanks also to Matt Gursky for pointing out to me the well-known, (but not to me at the time), splitting (0.13) during some conversations we had in Summer ’94.

I’m most grateful to Cliff Taubes for his careful reading of the manuscript and suggestions for its improvement, and for his continued support and interest in this program.

Finally, many thanks to Ramon del Castillo and especially my wife, Myong-hi Kim, for carrying out the huge task of converting the original paper into a TEXmanuscript.

1. Background Material.

§1.1. Throughout the paper, M will denote a compact, connected, oriented 3-manifold, without boundary. Let 𝕄 denote the space of smooth (C^{∞}) Riemannian metrics on M. This is an open cone, (the positive definite cone), in the linear space S^{2}(M) of smooth symmetric bilinear forms. For any g ∈ 𝕄, one has a natural identification

T_{g}𝕄 = S^{2}(M),  (1.1)

of the tangent space of 𝕄 at g. Let M denote the space of isometry classes of all C^{∞} Riemannian metrics; thus M is the quotient of 𝕄 by the action of the diffeomorphism group Diff(M) on 𝕄. It is well known, [Bes, Ch.4B], that the tangent space T_{g}𝕄 splits as

T_{g}𝕄 = Im δ^{∗} ⊕ Ker δ,  (1.2)

where δ : S^{2}(M) → Ω^{1}(M) is the divergence operator (w.r.t. g) on S^{2}(M), given by δ (α) = −(D_{e_{i}}α)(e_{i}, ·); here {e_{i}} is an orthonormal basis of T M. The operator δ^{∗} : Ω^{1}(M) → S^{2}(M) is the formal adjoint of δ, given by δ^{∗}(ω) = \frac{1}{2}L_{ω}g; L denotes the Lie derivative, and we are identifying vector fields and 1-forms via the metric. In the splitting (1.2), the factor Im δ^{∗} is the tangent space to the orbit of Diff(M) at g, while the factor Ker δ is naturally identified with the tangent space of the quotient M, at least when g has no isometries.

There is a natural (weak) Riemannian metric on 𝕄, the L^{2} metric, given for α, β ∈ T_{g}𝕄 by

⟨α, β⟩ = ∫_{M} tr_{g}(α ∘ β)dV_{g},  (1.3)

where α, β are considered as linear maps of T M, via g, and dV_{g} denotes the volume form of g. It is easily verified that this L^{2} metric is invariant under the action of Diff(M), and thus passes to the quotient to give the L^{2} metric on M. Further, the splitting (1.2) is orthogonal w.r.t. the L^{2} metric.

The subsets of 𝕄 and M consisting of metrics of volume 1 on M will be denoted by 𝕄_{1} and M_{1}. The discussion above is also valid for 𝕄_{1} and M_{1}.

The (weak) L^{2} Riemannian metric does not give a good (smooth) topology to 𝕄. For our purposes, it is natural to put the L^{2,2} topology on 𝕄, i.e. the topology given by the L^{2,2} Riemannian metric, defined on each T_{g}𝕄 by

||h||_{L^{2,2}} = ∫_{M} |h|^{2} + |Dh|^{2} + |D^{2}h|^{2} dV_{g} ^{1/2},  (1.4)

for h ∈ T_{g}𝕄. Here the norms | · |, and covariant derivative D are taken with respect to the metric g ∈ 𝕄. Thus, the L^{2,2} distance between two metrics g, g′ is the infimum of the lengths of smooth curves joining g and g′; the length of a curve γ is defined by

L(γ) = ∫ ||\frac{dγ}{dt} ||_{L^{2,2}(γ(t))} dt.  (1.5)

This norm corresponds formally to the Sobolev space L^{2,2} of functions with two weak derivatives in L^{2}. In fact, it is not difficult to verify, c.f. [E, p.21], that the L^{2,2} metric topology above induces the same topology as the L^{2,2} topology defined by local coordinates, i.e.

dist_{L^{2,2}}(g, g′) < ε ⇔ |g_{ij} − g′_{ij}|_{L^{2,2}} < ε′,  (1.6)

where | · |_{L^{2,2}} is the Sobolev topology on functions on bounded domains in ℝ^{3}, and the components g_{ij}, g′_{ij} are taken with respect to an arbitrary fixed coordinate atlas of M; here ε′ = ε′(ε) is small if ε is small, and vice versa. Although the two topologies defined by (1.5) and (1.6) are the same, the two metrics are very different in the large on 𝕄.

The Sobolev embedding theorem for L^{2,2} functions in dimension 3 reads

L^{2,2} ⊂ L^{1,6} ⊂ C^{1/2};  (1.7)

this is understood to apply to functions of compact support say in the unit ball in ℝ^{3}. Here C^{α} denotes the space of Hölder continuous functions with Hölder exponent α, L^{p,q} the Sobolev space of functions with p weak derivatives in L^{q}. Thus, at any g ∈ 𝕄, one has an estimate of the form

c_{s}(g)||α||_{C^{1/2}(g)} ≤ ||α||_{L^{2,2}(g)},  (1.8)

for any 2-tensor α. The Sobolev constant c_{s} depends strongly on the metric g, so that the estimate (1.8) is not uniform on 𝕄.

It is useful to compare the L^{2,2} metric with a somewhat weaker metric. Thus, define the T^{2,2} metric to be the Riemannian metric given on each T_{g}𝕄 by the norm

||h||_{T^{2,2}} = ∫_{M} |h|^{2} + |Dh|^{2} + |D^{∗}Dh|^{2} dV_{g} ^{1/2},  (1.9)

where D^{∗} is the L^{2} adjoint of D; (D^{∗}D is the so-called rough Laplacian of g). Since D^{∗}D is an elliptic operator, by elliptic regularity, there is a constant C = C(g) such that

||h||_{T^{2,2}} ≤ ||h||_{L^{2,2}} ≤ C(g) · ||h||_{T^{2,2}}.  (1.10)

Using (1.10), it can be shown that the topology defined by the T^{2,2} metric is the same as the topology defined by the L^{2,2} metric, c.f. again [E,p.21]. However the L^{2,2} and T^{2,2} metrics are far from being uniformly equivalent; the constant C, which by definition is the constant on which one has L^{2} elliptic estimates, c.f. [GT, Thm.8.8], cannot be chosen independent of g.

We return to the study of 𝕄 in more detail in §7.

§1.2. The solution to the Yamabe problem by Yamabe [Y], Trudinger [Tr] Aubin [Au1] and Schoen [Sc1], implies that in each conformal class [g] ⊂ 𝕄_{1} of metrics there is a preferred metric, namely a Yamabe metric , i.e. a metric of constant scalar curvature which minimizes the total scalar curvature functional S (0.1) restricted to the conformal class [g] ∩ 𝕄_{1}.

We let C denote the subset of 𝕄 consisting of all Yamabe metrics, and C_{1} the subset of unit volume Yamabe metrics; C is obtained from C_{1} by rescaling. If g is a fixed background metric in [g] ∩ 𝕄_{1} and ḡ is a Yamabe metric in [g] ∩ 𝕄_{1}, then one may write

ḡ = ψ^{4} · g,  (1.11)

where ψ is a smooth positive function on M, with L^{6} norm 1, w.r.t. dV_{g}, corresponding to volḡ M = vol_{g} M. The equation that ḡ have constant scalar curvature s̄ = inf S|_{[g]} is

ψ^{5} · s̄ = −8Δψ + ψ · s.  (1.12)

In general, although there always exists at least one Yamabe metric, it is unknown to what extent they are unique in their conformal class. It is well-known, and follows easily from the maximum principle applied to (1.12), that Yamabe metrics ḡ ∈ C_{1} are unique in their conformal classes, when the scalar curvature s̄ of ḡ, necessarily constant, is non-positive. In fact, in this case, ḡ is the unique metric of constant scalar curvature in [g] ∩ 𝕄_{1}. c.f. [Sc2].

Thus, in case σ(M) ≤ 0, any unit volume Yamabe metric has non-positive scalar curvature and so is unique. Further, it is well-known, c.f. [Bes, Cor.4.49], [Sc2] that if σ(M) ≤ 0, any metric of zero scalar curvature is flat, so that, with the exception of flat 3-manifolds, any Yamabe metric on M has negative scalar curvature. Conversely, if M is a flat 3-manifold, then M admits no metrics of positive scalar curvature, c.f. [Sc2], so that σ(M) = 0. Thus, on a flat 3-manifold, any flat metric realizes the Sigma constant (0.5).

On the other hand, if σ(M) > 0, i.e. M admits a metric g of positive scalar curvature, then there may be many (necessarily) positive constant scalar curvature metrics in [g] ∩ 𝕄_{1}.

We note that in case σ(M) ≤ 0, the space C of Yamabe metrics is an infinite dimensional submanifold of 𝕄. More generally, if g ∈ C, and −s_{g}/2 is not in the spectrum of the Laplacian, then an L^{2,2} neighborhood of g in C is an infinite dimensional submanifold of 𝕄; c.f. [Bes, Ch.4F], [Ks].

§1.3. As noted in the Introduction, the static vacuum Einstein equations

hr = D^{2}h, Δh = 0,  (1.13)

will play a fundamental role in the study of the degeneration of Yamabe metrics. Here D^{2}h is the Hessian of h, i.e. D∇h, where D is the covariant derivative and ∇ the gradient. The Laplacian will always be defined as Δh = tr D^{2}h, (sum of second derivatives), so that Δ has non-positive spectrum.

These are equations for a pair (g, h), defined on an open 3-manifold N. From the maximum principle, it is obvious that there are no (non-trivial) solutions of (1.13) on closed manifolds.

If one considers the 4-manifold X = N × S^{1}, with warped product metric g_{X} = g + h^{2}dθ^{2}, then the equations for the Ricci curvature r_{X} on (X, g_{X}) are

r_{X}(H, H) = r(H, H) − \frac{1}{h}D^{2}h(H, H), r_{X}(V, V) = −\frac{1}{h} · Δh · |V |^{2},  (1.14)

where H is tangent to N, V is tangent to S^{1}. c.f. [Bes, 9.106]. Thus the static vacuum Einstein equations (1.13) are exactly the equations that the Ricci curvature vanish on (X, g_{X}).

Using the regularity theory of elliptic equations, c.f. [GT, Ch.8], it is quite standard to prove that L^{2,2} weak solutions of (1.13) are C^{∞}, in fact real-analytic, in harmonic coordinates, in any domain on which h is bounded away from 0, (where the equations degenerate). Here one uses the fact that, to leading order, the Ricci curvature is given by the Laplacian of the metric in harmonic coordinates. Alternately, one may use regularity estimates for the Einstein metric (1.14) to prove smoothness of weak solutions, c.f. [DK].

Of course, a flat metric, with h an affine function, gives a trivial solution to the static vacuum equations. The canonical solution is given by the Schwarzschild metric (0.17). One has the following remarkable ‘black-hole uniqueness theorem’, proved by physicists.

Theorem 1.1. [I], [Ro], [BM]. Let (N, g, h) be a solution to the static vacuum equations (1.13), complete up to the locus Σ = {h = 0}, in the sense that the metric completion N̄ of (N, g) is given by N ∪Σ. Suppose further that the metric g extends smoothly to N ∪Σ, and that (N, g, h) is asymptotically flat, so that, outside a large compact set in N̄, the metric g is given in a chart by

g_{ij} = (1 + \frac{2m}{t}) δ_{ij} + γ_{ij},  h = 1 − \frac{m}{t} + O(t^{−2});  (1.15)

here, t = |x|, |D^{k}γ_{ij}| = O(t^{−2−k}) and m > 0.

If Σ is a compact, possibly disconnected surface, then (N, g, h) is the Schwarzschild metric of mass m for some m > 0.

It is shown in [An4] that the assumption that (N, g, h) is asymptotically flat is usually superfluous. Except in some rather rare situations, (which may occur however), it is a consequence of the assumptions that N ∪ Σ is complete and Σ is compact, (possibly singular).

The locus Σ = {h = 0} is called the event horizon in general relativity, and plays a special role. From the equations (1.13), note that if g is smooth up to Σ, then it follows immediately that D^{2}h = 0 on Σ. In particular, |∇h| = const. on each component of Σ, and it its straightforward to prove that |∇h| > 0 everywhere on Σ, c.f. [An4, Rmk.1.5]. Hence, each component of Σ is a totally geodesic surface, with |∇h| = const. > 0.

There are however many other solutions to the static vacuum equations (1.13). Usually, these are either singular at the event horizon Σ, or not complete away from Σ, or both. As indicated above the equations (1.13) may be viewed as an elliptic system, (in harmonic coordinates for example), away from Σ, but the equations formally degenerate at Σ.

We point out though that any weak L^{2,2} static vacuum solution on a domain D, with L^{2,2} potential function h, is smooth, in fact real-analytic, in the interior of D, without any assumptions that h is bounded away from 0 in D. Hence, in this case, Σ ∩ D is a smooth, totally geodesic surface, and g is smooth up to Σ, as above. Since no use will be made of this sharpening of the local regularity mentioned prior to Theorem 1.1, we do not include a proof, except to note that it is a straightforward exercise in elliptic regularity techniques, on the associated Ricci-flat 4-manifold.

A large and interesting class of explicit solutions of the static vacuum equations are given by the Weyl solutions, where (N, g) itself is a warped product of the form

N = V ×_{f} S^{1}, g = g_{V} + f^{2}dθ^{2},

and (V, g_{V}) is a Riemannian surface, c.f. [EK, §2.3-9], [Kr, Ch.16-18]; these metrics are discussed in much more detail in [An4]. For metrics of this form, the function

r = f · h

is harmonic on (V, g_{V}). Let z be the (locally defined) harmonic conjugate of r on V. Then the function

ν = log h  (1.16)

is an axially symmetric harmonic function on a domain in ℝ^{3}, where ℝ^{3} is given cylindrical coordinates (r, z, θ), θ ∈ [0, 2π]. The set I = {ν = −∞}, (considered as ∩_{n}ν^{−1}(−∞, n)), is usually assumed to be non-empty, (c.f. Theorem 3.2(I)), and in most circumstances, (but not always), is a subset of the axis A = {r = 0}. Note that the event horizon Σ corresponds to the locus I. The metric g is given in these cylindrical coordinates by

g = h^{−2}(e^{2λ}(dr^{2} + dz^{2}) + r^{2}dθ^{2}),  (1.17)

where λ is a solution to the integrability equations

λ_{r} = r(ν_{r}^{2} − ν_{z}^{2}), λ_{z} = 2rν_{r}ν_{z}.

Conversely, given any axially symmetric harmonic function ν on a domain in ℝ^{3}, the equations above determine λ (locally) up to a constant and the metric (1.17) gives a solution to the static vacuum equations with S^{1} symmetry.

Thus, Weyl solutions are completely determined locally by an axially symmetric harmonic function ν on a maximal domain D in ℝ^{3}.

There is a large variety of behaviors in the global geometry of Weyl solutions, c.f. [An4]. Probably the most interesting class of metrics are those for which the potential ν in (1.16) is a globally subharmonic function on ℝ^{3}. For these, one may use the value distribution theory of subharmonic functions on domains in ℝ^{3} to analyse the geometry of the associated Weyl metric.

If ν is subharmonic on ℝ^{3} and bounded above, say sup ν = 0, then the Riesz representation theorem c.f. [Ha, Thms. 3.9, 3.20], implies that ν may be globally represented as

ν(x) = − ∫_{ℝ^{3}} \frac{1}{|x − ξ|}dμ_{ξ},  (1.18)

where dμ_{ξ} is a positive Radon measure on ℝ^{3}, the Riesz measure of ν. For such axi-symmetric functions, one has the characterization

supp dμ = Ī ⇔ supp dμ ⊂ A.

The Riesz measure dμ in (1.18) encodes all the geometric information on the structure of such Weyl solutions. The fact that the Riesz measure is positive implies that such solutions have positive mass, in the sense of general relativity. Note that ν → sup ν = 0 as the distance to the event horizon goes to ∞.

On the other hand, there are solutions with negative mass. One may just take the function −ν in (1.18) for example, so that the potential is superharmonic. Note that in this case, u is unbounded above on supp dμ, while u tends to its infimum at large distances to supp dμ. Of course in this case, there is no event horizon.

It is worthwhile to list a few standard examples of Weyl solutions and their corresponding measures. In these examples, the metric is given globally by (1.17), and the mass is assumed positive.

From the point of view of the Riesz measure, perhaps the simplest example is the measure dμ given by a multiple of the Dirac measure at some point on A, so that ν = −m/t, t(x) = |x|, is a multiple of the Green’s function on ℝ^{3}. This gives rise to the Curzon solution [Kr, (18.4)],

g_{c} = e^{2m/t}(e^{−m^{2}r^{2}/t^{4}}(dr^{2} + dz^{2}) + r^{2}dθ^{2}).

The event horizon Σ corresponds formally to {0} and u = e^{−m/t}. This metric has a complicated singularity at the origin.

The Schwarzschild metric (0.17) is a Weyl metric, with measure dμ = \frac{1}{2}dA, where dA is the standard Lebesgue measure on a finite interval, say [−m, m] in A. This gives ν of the form

ν = \frac{1}{2} ln \frac{r_{+} + r_{-} − 2m}{r_{+} + r_{-} + 2m}, where r2_{±} = r^{2} + (z ± m)^{2}.

As mentioned before, the event horizon Σ here is a smooth totally geodesic 2-sphere of radius 2m.

It is easy to see that a Weyl solution (N, g) generated by a potential ν as in (1.18) for which supp dμ = Ī is a compact subset of the axis A, is asymptotically flat, in the sense of (1.15). Further, the simplest or most natural surfaces enclosing any finite number of compact components of Ī, and intersecting A outside Ī, are 2-spheres in N. Of course if supp dμ ⊂ A is non-compact, then the solution cannot be asymptotically flat.

Note that (sub)-harmonic functions of the form (1.18) form a convex cone. In particular, one thus has a natural linear superposition principle for Weyl solutions. For example, one may choose the measure dμ = \frac{1}{2}dA on two, or any number of disjoint intervals \{I_{j}\} on the axis A, provided the integral in (1.18) is finite. These correspond to solutions with ‘multiple black holes’. Although such solutions are essentially smooth up to the axis A, they do not define smooth Weyl solutions (N, g). There are cone singularities, (called struts in the physics literature), along geodesics (corresponding to A \ {I_{j}}) joining the 2-spheres of Σ, so that the metric g is not locally Euclidean along such curves. Nevertheless, the curvature of such metrics is uniformly bounded everywhere. Of course, the black hole uniqueness theorem, Theorem 1.1, also implies that such solutions cannot be smooth everywhere, when the number of intervals is finite.

It seems to be unknown whether there are any smooth Schwarzschild type metrics with infinitely many black holes, although it is natural to conjecture that such solutions do not exist, c.f. [An4].

On the other hand, there are Weyl solutions, c.f. [Sz], which are smooth and complete everywhere away from the event horizon Σ, for which Σ, (corresponding to I ⊂ A), consists of any number, including infinity, of components; here the Weyl metric is highly singular at the event horizon.

It is often useful in analysing the behavior of static vacuum solutions to consider the conformally equivalent metric

g̃ = u^{2} · g  (1.19)

on N, c.f. [EK, §2-3.5] and compare with (1.17). An easy calculation, c.f. [Bes, p.59] shows that the Ricci curvature r̃ of g̃ is given by

r̃ = 2(d log u)^{2} ≥ 0,  (1.20)

and

Δ̃ log u = 0.  (1.21)

This allows one to use methods and results on spaces with non-negative Ricci curvature and the behavior of harmonic functions on such spaces. Note that the metric g̃ is necessarily singular at the event horizon, even if the event horizon Σ is smooth in (N, g).

§1.4. We briefly summarize some of the main aspects of the theory of convergence/degeneration of metrics under uniform curvature bounds, but refer to the primary sources for further details.

Let V_{i} be a sequence of (possibly open) manifolds and let {γ_{i}} be a sequence of smooth Riemannian metrics on V_{i}. The sequence (V_{i}, γ_{i}) is said to converge to (V, γ) in the C^{1,α} topology, if first V_{i} is diffeomorphic to V, for all i sufficiently large, and second there exist diffeomorphisms ϕ_{i} : V → V_{i}, such that the pull-back metrics ϕ_{i}^{∗}γ_{i} converge to the metric γ in the C^{1,α} topology on V. This means that there is a smooth coordinate atlas on V for which the component functions of {ϕ_{i}^{∗}γ_{i}} converge to the component functions of γ; here the convergence is with respect to the usual C^{1,α} topology for functions defined on domains in ℝ^{3}. Note that this notion is well-defined for metrics γ which are not necessarily C^{∞}; for instance, it may well be that the limit metric γ is only a C^{1,α} metric on V.

In exactly the same way one defines convergence with respect to other topologies, for instance the weak or strong L^{2,2} topology. It follows from the Sobolev embedding theorem (1.7) that convergence in the strong L^{2,2} topology implies convergence in the L^{1,6} and C^{1/2} topologies. Further, convergence in the weak L^{2,2} topology implies convergence in the L^{1,p} and C^{α} topologies, for p < 6 and α < \frac{1}{2}.

We require the following definitions for a Riemannian 3-manifold (M, g), c.f. [An2, §3].

Definition 1.2. (I) The μ-volume radius at x is given by

ν(x) = sup \{r : \frac{vol(B_{y}(s) ∩ B_{x}(r))}{s^{3}} ≥ μω_{3}, for all y ∈ B_{x}(r), s ≤ r, \}  (1.22)

where ω_{3} is the volume of the unit ball in ℝ^{3}. The parameter μ is chosen to be an arbitrary but fixed small number, e.g. 10^{−2}, so that we will suppress the parameter μ.

We note that trivially one has the bound

ν(x) ≤ (μω_{3})^{−1} · vol M)^{1/3} .  (1.23)

(II). The L^{2,2} harmonic radius r_{h}(x), at x is the radius of the largest geodesic ball about x in which there exist harmonic coordinates u_{i} : B_{x}(r_{h}(x)) → ℝ, with respect to which the metric components g_{ij} satisfy

e^{−C}δ_{ij} ≤ g_{ij} ≤ e^{C}δ_{ij},  as bilinear forms,  (1.24)

and

r_{h}^{1/2}||∂^{2}g_{ij}||_{L^{2}(B_{x}(r_{h}(x)))} ≤ C.  (1.25)

(III). The L^{2} curvature radius ρ(x) is the radius of the largest geodesic ball at x such that for y ∈ B_{x}(ρ(x)), and D_{y}(s) = B_{x}(ρ(x)) ∩ B_{y}(s), one has the bound

\frac{s^{4}}{vol D_{y}(s)} ∫_{Dy(s)} |r|^{2} ≤ c_{o},  (1.26)

for any s ≤ ρ(x). Note that the left-hand side of (1.26) is not necessarily a monotonic function of s, for a fixed y, thus the need to vary the center point, as in (1.22). Define the harmonic radius r_{h}, or r_{h}(N) of (N, g) by r_{h}(N) = inf_{x∈N} r_{h}(x), and similarly for the L^{2} curvature radius ρ(N) of N. If N is a complete flat manifold, note that ρ(N) = ∞.

The constant C in (1.24)–(1.25) is an arbitrary but fixed parameter that may be taken to be 1. Similarly, c_{o} is a free parameter, but will be chosen to be a fixed sufficiently small number, say 10^{−3}, throughout the paper. Note that the bounds (1.24)-(1.26) are invariant under rescaling of the metric (one also rescales the coordinate functions u_{i}). In particular, these radii and the volume radius scale, i.e. behave under rescalings of the metric, as distance functions do. Observe from the definition that if y ∈ B_{x}(ρ(x)), then

ρ(y) ≥ dist(y, ∂B_{x}(ρ(x))).  (1.27)

The same estimate holds for r_{h}. Thus, ρ and r_{h} are Lipschitz functions, with Lipschitz constant 1. By the Sobolev embedding theorem, L^{2,2} ⊂ C^{1/2}, so that one also has C^{1/2} control of the metric components on B_{x}(r_{h}(x)).

It is important to note that the L^{2,2} harmonic radius and the L^{2} curvature radius are continuous with respect to convergence in the strong L^{2,2} topology. Thus, if (M_{i}, g_{i}) → (M, g) in the L^{2,2} topology, with x_{i} → x, then

lim_{i→∞} r_{h}(x_{i}, g_{i}) = r_{h}(x, g), and lim_{i→∞} ρ_{g_{i}}(x_{i}) = ρ_{g}(x).  (1.28)

This is more or less obvious for ρ and is proved for r_{h} in [An3, Lemma 2.2], c.f. also [AC,Prop.1.1]. However, (1.28) is not true if the convergence is only in the weak L^{2,2} topology.

There is an obvious relation between r_{h} and ρ, namely

ρ(x) ≥ c · r_{h}(x),  (1.29)

where c = c(C, c_{o}). To see this, we may assume (by rescaling) that r_{h}(x) = 1. The bound (1.24) then implies upper and lower bounds on vol B_{x}(1), while the bound (1.25) provides a bound on the L^{2} norm of curvature on B_{x}(1). This shows ρ(x) ≥ c, as required.

On the other hand, the opposite inequality, ρ(x) ≤ c · r_{h}(x), is not true, as seen for example on compact flat manifolds, or more generally on manifolds which are highly collapsed on the scale of their curvature, that is for which ν << ρ. On the other hand, under the presence of a lower bound on ν(x), one does obtain a bound of the form ρ(x) ≤ c · r_{h}(x), provided ρ(x) is not too large, i.e. ρ(x) ≤ K, where c = c(ν(x), K), c.f. [An2, §3],[An3].

The natural bounds on sequences of Yamabe metrics that we obtain are bounds on the L^{2} norm of (components of) the curvature. The following results, proved in [An2, §3], summarize the behavior of sequences of metrics on 3-manifolds, having a uniform L^{2} bound on curvature. These are extensions of the fundamental (L^{∞}) Cheeger-Gromov theory of convergence and collapse of Riemannian manifolds, see [C], [G], [CG1,2].

Theorem 1.3. Let {g_{i}} be a sequence of metrics in 𝕄_{1}(M), where M is a closed 3-manifold. Suppose there is a uniform bound

∫_{M} |r_{g_{i}}|^{2}dV_{i} ≤ Λ < ∞.  (1.30)

Then there is a subsequence, also called {g_{i}}, and diffeomorphisms ψ_{i} of M such that exactly one of the following occurs: (I) (Convergence) The metrics ψ_{i}^{∗}g_{i} converge in the weak L^{2,2} topology to a L^{2,2} metric g_{o} on M. (II) (Collapse) The metrics ψ_{i}^{∗}g_{i} collapse M along a sequence of orbit structures of a graph manifold

structure. In particular, M is necessarily a graph manifold. The metrics ψ_{i}^{∗}g_{i} collapse each orbit O_{x}, (namely a circle or torus), of a sequence of orbit structures to a point, as i → ∞, i.e. diam_{ψ_{i}^{∗}g_{i}} O_{x} → 0, ∀x ∈ M.

(III) (Cusps) There is a maximal open domain Ω, (not necessarily connected), such that the metrics ψ_{i}^{∗}g_{i} converge, uniformly on compact subsets in the weak L^{2,2} topology, to an L^{2,2} metric g_{o} defined on Ω, of volume ≤ 1. Any smooth compact domain K ⊂ Ω embeds in M and for sufficiently large K ⊂ Ω, the complement M \ K has the structure of a graph manifold, part of which is collapsed along a sequence of orbit structures as in II.

Graph manifolds are 3-manifolds which are unions of Seifert fibered spaces glued along toral boundary components; thus, they have naturally defined embedded circles or tori, see [CG1], [An1,2]. From the Sobolev embedding theorem, the convergence in cases I and III above is also in the L^{1,p} and C^{α} topologies, p < 6, α < \frac{1}{2}.

The distinction between these three cases is determined by the behavior of the volume radius ν_{i}. Case I occurs if ν_{i} is uniformly bounded below, ν_{i} ≥ ν_{o} ≥ 0, Case II occurs if ν_{i} → 0 everywhere on M, while in Case III, there are regions in (M, g_{i}) where ν_{i} is bounded below and regions where it goes to 0.

The following elementary result shows that {(M, g_{i})} degenerates in the sense of (0.8) only if ρ_{i}(x_{i}) → 0, for some x_{i} ∈ M.

Lemma 1.4. Let g be a unit volume metric on M for which there is a uniform lower bound on the L^{2} curvature radius ρ(M), say

ρ(M) ≥ ρ_{o}.  (1.31)

Then there is an explicit constant Λ = Λ(ρ_{o}) such that

∫_{M} |z|^{2}dV ≤ Λ.  (1.32)

Proof. From the definition of ρ, we have

∫_{B_{x}(ρ_{o})} |z|^{2}dV ≤ c_{o} \frac{vol B_{x}(ρ_{o})}{ρ_{o}^{4}},

for any geodesic ball B_{x}(ρ_{o}) ⊂ (M, g). Choose a maximal family of disjoint balls B_{x_{k}}, k = 1, ..., m of radius ρ_{o}/4 in (M, g); thus the corresponding balls of radius ρ_{o}/2 cover M. Now observe that there is a uniform upper bound N on the multiplicity of this covering, independent of ρ_{o}, M. In fact, since c_{o} in (1.26) is sufficiently small, (c_{o} = 10^{−3} suffices), the geometry of B(ρ_{o}/2) is very close to that of a Euclidean ball, (or its quotient by Euclidean isometries), so that the Besicovitch covering theorem holds, c.f. [M, Theorem 2.7]. Thus

∫_{M} |z|^{2}dV ≤ N ∑ ∫_{B_{x_{k}} ( \frac{ρo}{2} )} |z|^{2}dV ≤ \frac{Nc_{o}}{ρ_{o}^{4}} ∑ vol B_{x_{k}} (\frac{ρ_{o}}{2} ) ≤ C \frac{Nc_{o}}{ρ_{o}^{4}} ∑ vol B_{x_{k}} (\frac{ρ_{o}}{4} ) ≤ C \frac{Nc_{o}}{ρ_{o}^{4}} vol M.

■

There are also local versions of Theorem 1.3 which will be frequently used below. While there are related results which hold in the collapse case, c.f. [An2, §3], we consider only the non-collapsing situation here; c.f. [An2, Remark 3.6] for the following result.

Theorem 1.5. Let (U_{i}, g_{i}, x_{i}) be a pointed sequence of smooth Riemannian 3-manifolds such that

ρ_{i}(x_{i}) ≥ ρ_{o} > 0, ν_{i}(x_{i}) ≥ ν_{o} > 0, diam U_{i} ≤ D < ∞,

and

dist(x_{i}, ∂U_{i}) ≥ δ,

for some arbitrary positive constants ρ_{o}, ν_{o}, D, δ. Then for any given ε > 0 sufficiently small, there are smooth domains V_{i} ⊂ U_{i} with ε/2 ≤ dist(∂V_{i}, ∂U_{i}) ≤ ε, such that a subsequence of (V_{i}, g_{i}) converges, modulo diffeomorphisms, to a limit L^{2,2} Riemannian manifold (V, g_{o}). The convergence is in the weak L^{2,2} topology. In particular, the limit domain V embeds in the domains U_{i}.

In studying the degeneration of a sequence {g_{i}} of Yamabe metrics on M, we will blow-up the metrics in neighborhoods of points where the curvature of g_{i} goes to infinity, i.e. consider the behavior of the rescaled sequence

g′_{i} = ρ(x_{i})^{−2} · g_{i},  (1.33)

when ρ(x_{i}) → 0; of course, ρ(x_{i}) is the L^{2} curvature radius of g_{i} at x_{i}. Theorem 1.5 will be used to examine the behavior of this sequence. In considering limits of {g′_{i}}, one must always consider based limits, i.e. limits w.r.t. a sequence of base points; the points {x_{i}} will always be chosen to be the base points. Thus, assuming for instance that ν′(x) ≥ ν_{o}, for all x ∈ B′_{x_{i}}(1), for some ν_{o} ≥ 0, the pointed sequence {(B′_{x_{i}}(1), g′_{i}, x_{i})} has a subsequence converging weakly in L^{2,2} to a limit (B′, g′, x), x = lim x_{i}, with limit L^{2,2} metric g′. The convergence g′_{i} → g′ is also understood to be in the pointed Gromov-Hausdorff topology [G, Ch.5A].

Any smooth compact subdomain of B′ is naturally, but not canonically, embedded as a (very small) domain in (M, g_{i}) and the structure of the limit (B′, g′) mirrors the very small scale behavior of (M, g_{i}) near x_{i} as i → ∞.

Finally, some general remarks. Since we are constantly dealing with the behavior of sequences {g_{i}}, {g′_{i}} etc., we will often drop the subscript i, or prime, in order to simplify notation, when there is no danger of confusion. The main point is to establish uniform estimates, independent of i. Similarly, we will often pass to subsequences to obtain convergence, without always indicating the specific subsequence. A sequence {α_{i}} is said to sub-converge if a subsequence converges.

2. Initial Global Estimates for Yamabe Metrics

In this section, we derive a number of simple global relations on the behavior of the gradient of the (normalized) scalar curvature functional, i.e. the traceless Ricci curvature −z, as well various components of z, on the space of Yamabe metrics C.

For a given smooth metric g ∈ C, consider the operator L : S^{2}(M) → C^{∞}(M) giving the derivative or linearization of the scalar curvature function at g, i.e.

L(α) = s′(α) = \frac{d}{dt}s_{(g+tα)}.  (2.1)

This operator has been classically studied, for instance in general relativity, and is given, c.f. [Bes, Ch.1K], by

L(α) = −Δ tr α + δδα − ⟨r, α⟩,  (2.2)

The L^{2} adjoint of L, (w.r.t. the metric g), L^{∗} : C^{∞}(M) → S^{2}(M), is the expression

L^{∗}(h) = D^{2}h − Δh · g − h · r.  (2.3)

This is an (overdetermined) elliptic operator, and thus by general elliptic theory there is a splitting of T_{g}𝕄, orthogonal w.r.t. the L^{2} metric, of the form

T_{g}𝕄 = Im L^{∗} ⊕ Ker L.  (2.4)

To our knowledge, the splitting (2.4) first appeared in work of Berger-Ebin [BE], where it is attributed to Fadeev and Nirenberg, c.f. also [Bes, Ch.4F].

Of course one sees immediately that

−r = L^{∗}(1),  (2.5)

corresponding to the fact that the L^{2} gradient of the functional v^{−1} ∫ s dV, (at a Yamabe metric), is given by −r.

On the other hand, one may decompose the traceless Ricci tensor z (= z_{g}) with respect to this splitting and write

z = L^{∗}f + ξ,  (2.6)

where ξ ∈ Ker L and f is a smooth function on M. Clearly ξ and L^{∗}f are uniquely determined by z. If

Ker L^{∗} = {0},  (2.7)

then f is also uniquely determined by z.

To examine Ker L^{∗}, suppose h ∈ Ker L^{∗}, so that D^{2}h − Δh · g − h · r = 0. Taking the trace, one obtains

−2Δh − sh = 0,  (2.8)

i.e. h is an eigenfunction of the Laplacian, with eigenvalue −\frac{s}{2}. Suppose first that σ(M) ≤ 0. Then it follows, see §1.2, that s < 0, (unless M is a flat 3-manifold), and since the Laplacian has non-positive spectrum, the only solution of (2.8) is h = 0. In other words, σ(M) ≤ 0 implies Ker L^{∗} = 0, except in the special case that (M, g) is a flat 3-manifold, where Ker L^{∗} consists of the constant functions. In case σ(M) > 0, Ker L^{∗} may well be non-zero; see §3.6 and §6.4 for further discussion.

Remark 2.1. It is worth emphasizing at this point that although z is locally determined by g, neither L^{∗}f nor ξ in (2.6) are locally determined by g, (in contrast to the terms in (2.5)). They both depend on the global geometry of (M, g), in particular on the global volume or scale of (M, g). Thus they cannot be expressed locally in terms of the metric and its derivatives; see Theorem 2.10 for a simple concrete illustration of this.

On the other hand, in analogy to (2.5), note that z = H^{∗}(−1), where H^{∗}(f) = D^{2}f −Δf ·g −f ·z is the L^{2} adjoint of the operator H, H(α) = −Δ tr α+δδα −⟨z, α⟩, giving the derivative or linearization of the functional v^{2/3} · s on C.

Returning to the general discussion, we may add the equations (2.5) and (2.6); this gives the L^{2} splitting for the metric g, (considered as an element of T_{g}𝕄), i.e.

−\frac{s}{3} · g = L^{∗}(1 + f) + ξ,  (2.9)

so that, setting u = 1 + f gives

L^{∗}(u) + ξ = −\frac{s}{3} · g.  (2.10)

Note that at least in the case σ(M) ≤ 0, g is never in Im L^{∗} unless (M, g) is Einstein. To see this, suppose there is an h such that

L^{∗}h = g.

Taking the trace gives

2Δh + sh = −3.

It follows that h + \frac{3}{s} is an eigenfunction of Δ, with eigenvalue −\frac{s}{2}, so that the argument above implies (M, g) must be Einstein. The same reasoning shows that if σ(M) ≤ 0, then z ∈ Im L^{∗} only if (M, g) is Einstein; this is also conjectured to be the case when σ(M) > 0, c.f. [Bes, Remark 4.48].

Thus, the pair (f, ξ) measure in a certain global way the deviation from g being an Einstein metric on M.

Taking the L^{2} norm of both sides of (2.10) gives the following simple but important estimate.

Theorem 2.2. For f and ξ defined as above, and u = 1 + f, v = vol M, one has

∫ |L^{∗}u|^{2} + ∫ |ξ|^{2} = \frac{s^{2}}{3} · v.  (2.11)
■

Thus, one has apriori bounds on the L^{2} norms of L^{∗}(u) and ξ. Referring back to (2.6), it follows that the component of z in Ker L is apriori bounded in L^{2}. Thus, z is uncontrolled in L^{2}, say on a sequence g_{i} ∈ C_{1} with scalar curvature bounded below, only in the direction Im L^{∗}.

The splittings (2.6), (2.10) immediately give corresponding trace equations

2Δf + sf = tr ξ,  (2.12)

2Δu + su = tr ξ + s.  (2.13)

These equations easily lead to the following identities.

Lemma 2.3. For an arbitrary Yamabe metric g ∈ C, one has the relations

∫ |ξ|^{2} = −\frac{s}{3} ∫ tr ξ,  (2.14)

and

∫ tr ξ = s ∫ f.  (2.15)

Proof. For ξ ∈ Ker L, note by (2.2) that ∫ ⟨r, ξ⟩ = 0. Hence

∫ ⟨z, ξ⟩ = ∫ ⟨r, ξ⟩ − \frac{s}{3} ∫ ⟨g, ξ⟩ = −\frac{s}{3} ∫ tr ξ.

But also

∫ ⟨z, ξ⟩ = ∫ ⟨L^{∗}f, ξ⟩ + ∫ |ξ|^{2} = ∫ |ξ|^{2}.

This gives (2.14).

The equation (2.15) follows immediately from the trace equation (2.12) by integrating over M. ■

Of course (2.14) and (2.15) combine to give

∫ |ξ|^{2} = −\frac{s^{2}}{3} ∫ f.  (2.16)

Note that if g ∈ C_{1}, then (2.11) implies that the L^{2} norm of ξ is bounded by \frac{s^{2}}{3} .

Definition 2.4. For g ∈ C ∩ 𝕄_{1}, define the quantities δ ∈ [0, 1] and λ ∈ [0, 1] by

δ = − ∫_{M} f, λ = 1 − δ = ∫_{M} u.  (2.17)

The behavior of the quantities δ and λ plays a fundamental role in the discussion. For g ∈ C not of unit volume, δ, λ are defined by the averages in (2.17); thus, they are invariant under scaling.

One immediate consequence of this discussion is obtained by taking the L^{2} inner product of (2.10) with z, to give the interesting bound

∫ u|z|^{2} = ∫ ⟨ξ, z⟩ = ∫ |ξ|^{2} ≤ \frac{s^{2}}{3} · v.  (2.18)

In particular, if one had an estimate of the form u ≥ u_{o} > 0 on (M, g), then (2.18) gives an L^{2} bound on z; compare with §0.

Another important L^{2} orthogonal splitting of T_{g}𝕄 relevant to the study of the functional S|_{C} is given by

T_{g}𝕄 = T_{g}C ⊕ N_{g}C,  (2.19)

where

T_{g}C = {χ ∈ T_{g}𝕄 : L(χ) = const.}, N_{g}C = {τ ∈ T_{g}𝕄 : τ = L^{∗}(h), ∫_{M} h = 0} ,  (2.20)

as in (0.10). At least in the case where σ(M) ≤ 0, i.e. where Yamabe metrics are the unique metrics of constant scalar curvature in their conformal class, the spaces in (2.19) are the tangent and normal spaces to the space of Yamabe metrics C. In fact, using the splitting (2.19), one may show that C is an infinite dimensional submanifold of 𝕄, c.f. [Bes, Ch.4F], [Ks]. In case however −s/2 ∈ Spec (Δ) , the spaces T_{g}C and N_{g}C should only be considered as formal tangent spaces to C. (Recall that Δ has non-positive spectrum).

Note that the splittings (2.4) and (2.19) differ by only 1-dimensional factors. We may then decompose z also with respect to this splitting, and write

z = z^{T} + z^{N}.  (2.21)

The component −z^{T} , more precisely −v^{−1/3} · z^{T} , is the L^{2} gradient of the functional

v^{2/3} · s : C → ℝ,  (2.22)

(again assuming −s/2 ∉ Spec (Δ)). One sees this by observing that −v^{−1/3} ·z is the L^{2} gradient of the (volume normalized) total scalar curvature functional S : 𝕄 → ℝ in (0.1) and z^{T} is the L^{2} projection of z onto C.

Now the space T C may be further decomposed into

T C = T L ⊕ NL = Ker L ⊕ (Im L^{∗} ∩ T C),  (2.23)

where

T L = {ξ ∈ T C : L(ξ) = 0}, NL = {ϕ ∈ T C : ϕ = L^{∗}(h), some h}.  (2.24)

Clearly, the factor NL = (Im L^{∗} ∩ T C) is 1-dimensional.

Observe that T L corresponds to the tangent space to the level sets L of the functional s = S|_{C} while NL corresponds to the normal space of L in T C. The level sets L of s and the level sets H of v^{2/3} · s are obviously hypersurfaces in C at non-critical points of these functionals. These of course coincide on C ∩ 𝕄_{1}, but do not coincide off 𝕄_{1}. Note also that by definition, z^{T} is L^{2} orthogonal to the tangent spaces of H.

The splittings (2.4), (2.19) and (2.23) are all compatible with the splitting (1.2), since constant scalar curvature is a diffeomorphism invariant, see also [BE].

In particular, the L^{2} projection of z, z^{T} ∈ T C can be split further as

z^{T} = L^{∗}k + ξ, where L^{∗}k ∈ NL, ξ ∈ T L.  (2.25)

Lemma 2.5. The function k is characterized by the basic property that

LL^{∗}k = const. = −\frac{1}{v} ∫ |z^{T} |^{2}.  (2.26)

Proof. Apply the operator L to both sides of (2.25) to obtain

L(z^{T}) = LL^{∗}k,

since L(ξ) = 0. Since z^{T} ∈ T C, L(z^{T}) is a constant function and

∫ L(z^{T})dV = ∫ ⟨L^{∗}1, z^{T} ⟩dV = − ∫ ⟨r, z^{T} ⟩dV =

= − ∫ |z^{T} |^{2}dV − ∫ ⟨z^{T} , z^{N}⟩dV − \frac{s}{3} ∫ tr z^{T} dV.

The components z^{T} and z^{N} are L^{2} orthogonal. Also ∫ tr z^{T} is the derivative of the functional S|_{C} in the direction of the metric g, i.e. its derivative under homothetic changes of the metric. Since the functional is scale-invariant, this derivative is 0, and the result follows.

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A straightforward calculation shows that the operator LL^{∗} is given by

LL^{∗}v = 2ΔΔv + 2sΔv − ⟨D^{2}v, r⟩ + v|r|^{2}.  (2.27)

Clearly, when Ker L^{∗} = {0}, the space of functions ϕ such that LL^{∗}ϕ = c, for some constant c, is 1-dimensional. This gives the following relation between the functions u and k.

Proposition 2.6. For a Yamabe metric g ∈ C, with s_{g} < 0, one has the relation

\frac{u}{λ} = −\frac{k}{δ} .  (2.28)

In general, the relation (2.28) holds mod Ker L^{∗}.

Proof. We have

LL^{∗}u = L (−\frac{s}{3} · g − ξ) = −\frac{s}{3}L(g) = \frac{s^{2}}{3} ;  (2.29)

the last equality just corresponds to the fact that varying the scalar curvature s in the direction of g, i.e. by a homothety, just changes s by a constant. Thus, by (2.26) and (2.29), both LL^{∗}k = α and LL^{∗}u = β are constant. It follows that

LL^{∗} (\frac{k}{α} − \frac{u}{β} ) = 0.

If Ker L^{∗} = 0, we see that

\frac{u}{β} = \frac{k}{α},

so that α and β are the mean values of k and u respectively. From (2.6) and (2.25), we have

z^{N} = L^{∗}(f − k),  (2.30)

so that, in particular, since z^{N} ∈ NC,

∫ f = ∫ k = −δ.  (2.31)

This implies (2.28).

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Using (2.29), it is easy to see that the function u minimizes the L^{2} norm of Im L^{∗} among functions with the same mean value, i.e.

Proposition 2.7. The function u/λ, (or k/δ), is characterized uniquely by the fact that

∫ |L^{∗}(u/λ)|^{2}dV ≤ ∫ |L^{∗}ϕ|^{2}dV,  (2.32)

for all functions ϕ on (M, g) with mean value 1.

Proof. Write

∫ |L^{∗}ϕ|^{2}dV = ∫ |L^{∗}(ϕ − u/λ) + L^{∗}(u/λ)|^{2}dV =
= ∫ |L^{∗}(ϕ − u/λ)|^{2}dV + ∫ |L^{∗}(u/λ)|^{2}dV − 2 ∫ ⟨L^{∗}(ϕ − u/λ), L^{∗}(u/λ)⟩dV.

But

∫ ⟨L^{∗}(ϕ − u/λ), L^{∗}(u/λ)⟩dV = ∫ ⟨(ϕ − u/λ), LL^{∗}(u/λ)⟩dV = 0,

where the last equality follows from the fact that LL^{∗}u is constant by (2.29), and ϕ and u/λ have the same mean value.

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The equation (2.28) contains the basic relation between the functions f and k, relating z and z^{T} . It is useful to derive several further relations.

Lemma 2.8. The following identities hold for any Yamabe metric (M, g):

ξ + \frac{s}{3} · g = λ(z^{T} + \frac{s}{3} · g),  (2.33)
∫ |L^{∗}k|^{2} = \frac{s^{2}}{3} \frac{δ^{2}}{λ} · v,  (2.34)

∫ |z^{T} |^{2} = \frac{s^{2}}{3} \frac{δ}{λ} · v.  (2.35)

Proof. First, recall that

L^{∗}u = −ξ − \frac{s}{3} · g and L^{∗}k = z^{T} − ξ.

From (2.28), one then obtains

\frac{δ − 1}{δ}z^{T} = \frac{δ − 1}{δ}ξ − ξ − \frac{s}{3} · g = −\frac{1}{δ}ξ − \frac{s}{3} · g,

so that from Definition 2.4,

ξ + \frac{δ · s}{3} · g = λz^{T} .  (2.36)

which implies (2.33). Thus the tensors ξ + \frac{s}{3}g and z^{T} + \frac{s}{3}g are proportional. If ξ_{o} and z_{o}^{T} denote the trace-free parts of ξ and z^{T} , i.e. ξ_{o} = ξ − \frac{trξ}{3} · g, then one also has from (2.33) that

ξ_{o} = λ · z_{o}^{T} .

Next, from (2.28) and (2.29), we find

LL^{∗}k = −\frac{s^{2}}{3} \frac{δ}{λ},  (2.37)

Multiply (2.37) by k and integrate over M, using (2.31) to obtain (2.34). Finally from (2.16), we have ∫ |ξ|^{2} = \frac{s^{2}}{3} δ · v, so that (2.25) and (2.34) give (2.35).

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We now turn to the trace equation (2.12) in order to obtain estimates on f, or u.

Proposition 2.9. For any unit volume Yamabe metric g ∈ C_{1}, with s_{g} < 0, one has the estimate

||f||_{T^{2,2}} ≤ C,  (2.38)

where C is a constant depending only on |s_{g}|.

Proof. Recall from (1.9), that the T^{2,2} norm is given by

||f||_{T^{2,2}}^{2} = ||f||_{L^{2}}^{2} + ||df||_{L^{2}}^{2} + ||Δf||_{L^{2}}^{2}.

Multiplying (2.12) by f and integrating gives

\begin{aligned}2 ∫ |df|^{2} − s ∫ f^{2} & = − ∫ f · tr ξ \\ & ≤ \frac{1}{2}|s| ∫ f^{2} + \frac{1}{2}|s|^{−1} ∫ (tr ξ)^{2}, \end{aligned}  (2.39)

so that, since s < 0,

2 ∫ |df|^{2} + \frac{1}{2}|s| ∫ f^{2} ≤ \frac{1}{2}|s|^{−1} ∫ (tr ξ)^{2} ≤ \frac{1}{2}|s|δ,  (2.40)

where the last inequality follows from (2.16). Thus, one has an apriori bound

||f||_{L^{1,2}}^{2} ≤ C = C(|s|).

Returning to the trace equation (2.12), it also then follows that

||Δf||_{L^{2}}^{2} ≤ C = C(|s|).
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We note that the estimate (2.38) is false in case s_{g} > 0, see §3.6 for further discussion, while for s_{g} = 0, it is borderline. Namely if g is a Yamabe metric with s_{g} = 0, then either σ(M) > 0 or σ(M) = 0. In the former case, it is obvious that f ≡ −1 and ξ = 0 is one solution of the z-splitting equation (2.6) and hence it is the only solution since Ker L^{∗} = 0 when s_{g} = 0. Thus, in this case we have u ≡ 0. The splittings (2.6), (2.10) contain no essential information in this situation; they are equivalent to the identity (2.5). If however σ(M) = 0, then g realizes the Sigma constant on M, and thus g is Einstein, with r = 0. Thus, the solution to (2.6) is given by f = const. and ξ = 0, where the constant for f is however undetermined.

The bound (2.38) implies that there is an apriori bound for f in the T^{2,2} norm, in case σ(M) ≤ 0. On the other hand, we will see later in §3 that it is far from true that one has uniform bounds for ||D^{2}f||_{L^{2}} , i.e. uniform bounds for the L^{2,2} norm of f. This will be the case when one has non-flat blow-up limits. Thus, one has a breakdown of (uniform) elliptic regularity on {g_{i}}, c.f. (1.10).

We refer to §4.2 for some further discussion on the L^{2} behavior of u or f.

Next, consider the L^{2} bound on z^{T} = L^{∗}k + ξ. From Theorem 2.2, ξ is uniformly controlled in L^{2}, while k is a function in the 1-dimensional space Im L^{∗} ∩ T C. Thus, one would expect to be able to control z^{T} in a natural way. This is given by the following general result, which illustrates in a simple manner the highly global nature of the L^{2} projection operator onto T C.

Theorem 2.10. Let g be a unit volume Yamabe metric on M. Suppose there is a point x ∈ M, and arbitrary but fixed constants ρ_{o}, ν_{o} ≥ 0 such that

ρ(x) ≥ ρ_{o}, ν(x) ≥ ν_{o}.  (2.41)

Then there is a constant K = K(ρ_{o}, ν_{o}) such that

∫_{M} |z^{T} |^{2}dV_{g} ≤ K.  (2.42)

Proof. Since z^{T} ∈ T_{g}C is the L^{2} orthogonal projection of z onto T C, we have

∫_{M} |z^{T} |^{2}dV_{g} = inf ; ∫ ϕ=0 ∫_{M} |z − L^{∗}ϕ|^{2}dV_{g},  (2.43)

where the infimum is over all smooth functions ϕ on M with 0 mean value on (M, g); recall NC = Im L^{∗}ϕ, over mean value 0 functions on (M, g).

A straightforward computation gives

∫_{M} |z − L^{∗}ϕ|^{2}dV_{g} = ∫_{M}\{(1 + ϕ)^{2}|z|^{2} + |D^{2}ϕ|^{2} + (Δϕ)^{2} − \frac{2}{3}s|dϕ|^{2} + 2z(dϕ, dϕ) + \frac{s^{2}}{3} ϕ^{2}\}dV_{g}(2.44)

= ∫_{M}\{(1 + ϕ)^{2}|z|^{2} − |D^{2}ϕ|^{2} + 3(Δϕ)^{2} − \frac{4}{3}s|dϕ|^{2} + \frac{s^{2}}{3} ϕ^{2}\}dV_{g},

where the last inequality follows from the use of the Bochner-Lichnerowicz formula to eliminate the term z(dϕ, dϕ).

Given x ∈ M satisfying (2.41), choose a function ϕ, with ϕ ≥ −1 everywhere on M, ϕ ≡ −1 on M \B, where B = B_{x}(ρ_{o}/2) and with 0 mean value on (M, g). Note that since vol B ≥ c · min((ρ_{o}, ν_{o}))^{3}, so that B has a definite proportion of the volume of (M, g), one may choose such a ϕ so that sup ϕ ≤ H, where H depends only on ρ_{o}, ν_{o}. Now in the ball B the geometry of g is controlled in the L^{2,2} topology, see the discussion concerning (1.29). It follows that one may choose ϕ so that the L^{2,2} norm of ϕ is also uniformly controlled in B.

For such a choice of ϕ, it is clear that the expression on the right in (2.44) is uniformly controlled by ρ_{o} and ν_{o}. The estimate (2.42) then follows immediately from (2.44).

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Remark 2.11. In a related vein, suppose {g_{i}} is a sequence of unit volume Yamabe metrics on M, and {x_{i}} is a sequence of points in M satisfying (2.39). Then the local estimate

∫_{B_{i}} |z^{T} |^{2}dV_{g_{i}} → 0 as i → ∞,  (2.45)

implies the global estimate

∫_{M} |z^{T} |^{2}dV_{g_{i}} → 0 as i → ∞,  (2.46)

where B_{i} = B_{x_{i}}(ρ_{o}). To see this, from (2.25) and (2.26), we have L(z^{T}) = −||z^{T} ||_{L^{2}}. Multiplying this by a suitable cutoff function supported in B_{i} implies

vol_{g_{i}} B_{i} · ||z^{T} ||_{L^{2}} ≤ c ∫ ⟨L^{∗}, z^{T} ⟩ ≤ c ∫_{B_{i}} |z^{T} |^{2},

which gives the result.

Remark 2.12. From (2.35), we see that in case σ(M) ≠ 0, a bound on ||z^{T} ||_{L^{2}} is equivalent to a bound on λ away from 0. More precisely, if {g_{i}} is a sequence of unit volume Yamabe metrics with scalar curvature bounded away from 0 (and −∞), then ||z^{T} ||_{L^{2}} remains uniformly bounded exactly when λ, the mean value of u, remains bounded away from 0.

On the other hand, this is certainly not the case when σ(M) = 0 or more generally when s_{g_{i}} → 0. Suppose for instance (M, g_{o}) is a flat 3-manifold, so that g_{o} realizes σ(M). Let g_{t} be a smooth curve of unit volume Yamabe metrics on M through g_{o}, and let α = \frac{dg_{t}}{dt} satisfy ||α||_{L^{2}(g_{t})} = 1. We have

\frac{d}{dt}s_{g_{t}} = ∫_{M}⟨z^{T} , α⟩ ≤ (∫_{M} |z^{T} |^{2})^{1/2} ,

so that

(\frac{d}{dt}s_{g_{t}})^{2} ≤ ∫_{M} |z_{g_{t}}^{T}|^{2}.

Since s_{g_{t}} is a smooth function of t, with s_{g_{o}} = 0, the maximal value, it is clear that

−s_{g_{t}} << −\frac{d}{dt}s_{g_{t}},

as t → 0. Thus, it follows that

\frac{1}{s^{2}} ∫_{M} |z^{T} |^{2}dV → ∞ as t → 0.  (2.47)

By (2.35), this implies that

λ → 0,  (2.48)

as t → 0. Further, by the proof of Proposition 2.8, |∇u| → 0 in L^{2} and since u converges smoothly to its limit here, we have

u → 0 in C^{o}.

Note that in this example, one still has of course ||z^{T} ||_{L^{2}} ≤ C, in fact ||z^{T} ||_{L^{2}} → 0 as t → 0.

3. Existence of Non-Flat Blow-Ups.

In this section, we will analyse the degeneration of sequences of Yamabe metrics using the structural results in §2. We recall (Lemma 1.4), that a sequence of unit volume Yamabe metrics {g_{i}} degenerates, in the sense that the curvature becomes unbounded in L^{2}, only if the L^{2} curvature radius ρ_{i}(M) of (M, g_{i}) goes to 0.

§3.1. As indicated in §0, the static vacuum Einstein equations

ur = D^{2}u, Δu = 0,  (3.1)

play the fundamental role in the understanding of degenerations of sequences of Yamabe metrics. It is immediate from the definition that these equations are equivalent to the equation

L^{∗}u = 0,  (3.2)

on scalar-flat manifolds. Obviously, there are no non-trivial solutions of the equations (3.1) on closed manifolds, although (3.1) or (3.2) may have locally defined solutions.

We discuss briefly the relation between the equation (3.2) and Einstein metrics on 4-manifolds, since it does not seem to appear in the literature.

Proposition 3.0. Let g be a metric of constant scalar curvature on a 3-dimensional domain Ω, with smooth boundary ∂Ω. Then any solution of (3.2) on Ω with u > 0 on Ω gives an Einstein 4-manifold of the form

X^{4} = Ω ×_{u} S^{1},  (3.3)

with scalar curvature s_{X} = 2s_{g}.

Proof. Let u > 0 be a solution to (3.2) in Ω. For any form η ∈ T_{g}𝕄 of compact support in Ω, we then have

∫ u · L(η) dV_{g} = ∫ ⟨L^{∗}u, η⟩dV_{g} = 0.

Now consider the warped product X^{4} = Ω ×_{u} S^{1}, with metric given by

g_{X} = g + u^{2}dθ^{2}.  (3.4)

The volume form of g_{X} is given by udV_{g} ∧ dθ, and one computes, c.f. [Bes, Ch.9J], that the scalar curvature s_{X} of g_{X} is given by

s_{X} = s − 2\frac{Δu}{u}.  (3.5)

Thus, applying the divergence theorem

∫_{X} s_{X}dV_{X} = ∫_{Ω} (s − 2\frac{Δu}{u}) u dV_{g} = ∫_{Ω} u · s dV_{g} − 2 ∫_{∂Ω}⟨du, ν⟩dA_{g},

where ν is the outward unit normal.

Consider a compactly supported 1-parameter variation of g_{X}, i.e. a curve of metrics g_{X}(t) on X, with g_{X}(0) = g_{X}, of the form g_{X}(t) = g_{t} + u_{t}^{2}dθ^{2}. We suppose also that vol_{g_{X}(t)}(X) = vol_{g_{X}} (X). Then, for α = \frac{d}{dt}g_{X}(t)|_{t=0}, we have at t = 0,

\frac{d}{dt} ∫_{X} s_{X}(t) dV_{t} = \frac{d}{dt} ∫_{Ω} s_{t}u_{t} dV_{t} − 2\frac{d}{dt} ∫_{∂Ω}⟨du_{t}, ν⟩_{g_{t}} dA_{g_{t}}.

Now the boundary term vanishes, since the variation is of compact support. For the first integral, we have

\frac{d}{dt} ∫_{Ω} s_{t}u_{t}dV_{t} = ∫_{Ω} s′udV + s ∫_{Ω}(udV)′,

where we have used the fact that s is constant. The second integral here vanishes, since the variation is volume preserving, while for the first integral

∫_{Ω} s′udV = ∫_{Ω} L(α)udV = ∫_{Ω}⟨L^{∗}u, α⟩dV = 0.  (3.6)

Thus the gradient ∇S of the unnormalized scalar curvature functional on metrics on X vanishes when paired with all compactly supported, volume preserving, variations for which S^{1} acts by isometries. By the so-called symmetric criticality principle, c.f. [Bes, Thm.4.23], it follows that ∇S = 0 on (X, g_{X}), i.e. g_{X} is an Einstein metric. The scalar curvature s_{X} is determined by (3.5).

Alternately, the equations (1.14) can be used to show that the equation (3.2) is equivalent to the condition that g_{X} is Einstein.

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Of course a similar relation is valid in all dimensions. Conversely, the proof shows that if (X, g_{X}) is an Einstein 4-manifold of the form (3.4), then (3.2) holds. We note that the Einstein metric on X may (or may not) have singularities if Ω includes points where u = 0.

The equations (1.14) for the Ricci curvature of the 4-manifold (X, g_{X}), together with the basic identity (2.10), show that the Ricci curvature r_{X}, or the Einstein tensor r_{X} − \frac{1}{2}s_{X}g_{X}, of X may be expressed solely in terms of ξ, s and u. In the language of general relativity, (X, g_{X}) may be viewed as a (Riemannian) space-time with a matter or field term involving only ξ, s and u. In regions where u is bounded away from 0 and ∞, this term is apriori bounded in L^{2}.

§3.2. Although (3.1) has no non-trivial global solutions on compact manifolds, it is closely related to local degenerations of Yamabe metrics. Thus, suppose {x_{i}} is a sequence in (M, g_{i}) such that

ρ_{i}(x_{i}) → 0.  (3.7)

Consider the rescaled metrics

g′_{i} = ρ_{i}(x_{i})^{−2} · g_{i}.  (3.8)

Throughout §3, we make the following (weak) non-collapse assumption on sequences {g_{i}} of Yamabe metrics: for some arbitrary, but fixed ν_{o} > 0,

ν_{i}(x) ≥ ν_{o} · ρ_{i}(x).  (3.9)

The inequality (3.9) is scale-invariant, and it implies that in the scale where ρ(x) ∼ 1, as in (3.8), that one has a uniform lower bound on the volume of geodesic r-balls, for r ≤ 1, in terms of Euclidean volumes; see §4.1 for some further remarks on the collapse case.

In particular, for the pointed sequence (M, g′_{i}, x_{i}) above, it follows from Theorem 1.5 that a subsequence of {g′_{i}} converges, modulo diffeomorphisms, in the weak L^{2,2} topology to a limit metric g′, defined on a unit ball B′_{x}(1), centered at x = lim x_{i}.

The following Proposition proves Theorem A(I).

Proposition 3.1. Let (M, g_{i}, x_{i}) be a pointed sequence of Yamabe metrics on a closed 3-manifold M satisfying (3.7) and (3.9) with s_{g_{i}} ≥ −s_{o}, for some s_{o} < ∞. Then the blow-up limit (B′_{x}(1), g′, x) is an L^{2,2} weak solution of the static vacuum Einstein equations (3.1).

Proof. Each metric g_{i} has an associated splitting (2.10), i.e.

L^{∗}u + ξ = −\frac{s}{3} · g,  (3.10)

where we have dropped the subscript i. Such a splitting holds also for the metrics g′_{i} in (3.8), with the same function u = u_{i}, since u is scale invariant. The scaling properties of curvature imply

s′_{i} = ρ_{i}(x_{i})^{2} · s_{i} → 0,  (3.11)

since s_{i} is uniformly bounded, (c.f. (0.4)), and ρ_{i}(x_{i}) → 0, by (3.7). Further, since ξ scales as curvature,

∫_{M} |ξ′_{i}|^{2}dV′_{i} = ρ_{i}(x_{i}) · ∫_{M} |ξ_{i}|^{2}dV_{i} → 0,  (3.12)

since ∫_{M} |ξ_{i}|^{2}dV_{i} is uniformly bounded, see (2.11). We emphasize that (3.12) uses in an essential way that M is 3-dimensional.

Thus, it follows from (3.10)-(3.12) that

(L′)^{∗}u_{i} → 0, strongly in L^{2}(M, g′_{i}),  (3.13)

and hence the limit L^{2,2} metric g′ satisfies

(L′)^{∗}u = 0.  (3.14)

in B′_{x}(1). Here, to be precise, we need to examine the limiting behavior of {u_{i}}. Suppose first that {u_{i}} is bounded in L^{2}(B_{i}), B_{i} = B′_{x_{i}}(1). Since then Δu_{i} → 0 in L^{2}(M, g′_{i}), it follows from standard elliptic estimates, c.f. [GT, Thm.8.8], that {u_{i}} is uniformly bounded in L^{2,2} in B_{i}(1 − δ) = B′_{x_{i}}(1 − δ), for any given δ > 0. Hence {u_{i}} sub-converges to a limit function u ∈ (L^{2,2})_{loc} on B′_{x}(1) and (3.14) holds weakly, i.e. when paired with smooth 2-tensors of compact support in B′_{x}(1).

If instead ||u_{i}||_{L^{2}(B_{i})} → ∞ as i → ∞, we just renormalize u_{i} by setting ū_{i} = u_{i}/||u_{i}||_{L^{2}(B_{i})}. When renormalizing (3.10) by the same factor, all the terms become even smaller and the argument proceeds as before. This renormalization process will recur several times throughout §3.

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From the discussion in §1.3, weak L^{2,2} solutions of the static vacuum equations are smooth, (at least in regions where the potential function u does not vanish).

The following examples of static vacuum solutions, although trivial, are important in understanding the structure of the arguments to follow.

Examples of Static Vacuum Solutions:

Super-trivial solutions (N, g, u): u ≡ 0, (N, g) arbitrary.

Trivial solutions: (N, g, u) = (ℝ^{3}, g_{o}, u_{o}), where g_{o} is a flat metric on ℝ^{3} and u_{o} is a constant or affine function. Similarly, one may have such solutions on flat quotients of ℝ^{3}.

Note that super-trivial solutions give no information whatsoever about the Riemannian manifold (N, g). Thus, in order for Proposition 3.1 to be of any use, one must study the sequence (M, g_{i}) of Yamabe metrics away from the locus where u_{i} approaches 0, see also Remark 3.15(i).

We have the following characterization of the trivial or flat solution, generalizing a classical result of Lichnerowicz [Li], (which assumes that u is asymptotically constant).

Theorem 3.2. (I) Let (N, g, u) be a complete solution to the static vacuum equations (3.1), i.e. (N, g) is a complete Riemannian manifold. If u > 0 on N, then N is flat, and u is constant.

(II) Let (N, g, u) be a solution of (3.1) and let U ⊂ N be any domain with smooth boundary on which u > 0. If t(x) = dist_{N}(x, ∂U), for x ∈ U, then there is an absolute constant K < ∞ such that

|z|(x) ≤ \frac{K}{t^{2}(x)}, and (u^{−1}|∇u|)(x) ≤ \frac{K}{t(x)}.  (3.16)

The constant K does not depend on the domain U, (provided u > 0 on U), or on the static vacuum solution (N, g).

Proof. The proof is deferred to the Appendix, since the methods do not bear directly on the main discussion to follow. (The proof of (3.16) is similar though to the proof of Theorem 3.3 below).

The relation between the non-existence (of non-trivial solutions) in (I) is closely related to the existence of the pointwise curvature estimate (3.16) in (II). This situation occurs frequently in geometric P.D.E.’s and statements of the form (I) and (II) are often equivalent. Thus, given (I), one obtains (II) by a basically standard scaling argument. Conversely, (II) immediately implies (I) since the function t ≡ ∞ in this case.

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As noted before, the canonical solution of the static vacuum equations (3.1) is the Schwarzschild metric (0.17). This metric is characterized by the conditions in Theorem 1.1.

§3.3. In this subsection, we show that the curvature is controlled in L^{2} in regions of (M, g) where the level sets of u = 1 + f do not come too close together. Let

L^{c} = {x ∈ M : u(x) = c} and U^{c} = {x ∈ M : u(x) > c},

denote the c-level and super-level sets of u.

Theorem 3.3. Let g be a Yamabe metric on a closed 3-manifold M, of volume 1, satisfying s_{g} ≥ −s_{o} > −∞ and (3.9), i.e. ν(x) ≥ ν_{o} · ρ(x). Given any constant c > 0, there is a constant ρ_{o} = ρ_{o}(c, s_{o}, ν_{o}) > 0, such that, for any x ∈ U^{c},

ρ(x) ≥ ρ_{o} · min{1, dist(x, L^{c})}.  (3.17)

Proof. Note that the estimate (3.17) is exactly the L^{2} analogue of the estimate (3.16), but pertains to quite general Yamabe metrics while (3.16) holds only for the much more rigid class of static vacuum solutions. The proof of (3.17) reduces to that of (3.16) or Theorem 3.1(I) by taking blow-up limits.

Thus, we assume (3.17) is false, and will derive a contradiction. Given c > 0, s_{o} < ∞, if (3.17) does not hold, then there is a sequence of Yamabe metrics γ_{k} on M_{k} such that vol_{γ_{k}} M_{k} = 1, s_{γ_{k}} ≥ −s_{o}, and

\frac{ρ_{k}(x_{k})}{min\{v^{1/3}, dist_{γ_{k}}(x_{k}, L^{c})\}} → 0, as k → ∞,  (3.18)

for some sequence of points x_{k} ∈ U^{c} = U^{c}(k); here, we replace the constant 1 in (3.17) by v^{1/3} = vol_{γ_{k}} M_{k}^{1/3} = 1, so that the equation (3.18) is scale invariant. Choose points y_{k} realizing the minimum of the ratio in (3.18); it follows that ρ_{k}(y_{k}) → 0, as k → ∞. We rescale the metrics to make ρ_{k}(y_{k}) of size 1. Thus, define metrics γ′_{k} = ρ_{k}(y_{k})^{−2}γ_{k} and consider the sequence of pointed Riemannian manifolds (U^{c}, γ′_{k}, y_{k}). By construction

ρ′_{k}(y_{k}) = 1,  (3.19)

and

vol_{γ′_{k}} M_{k} → ∞, dist_{γ′_{k}}(y_{k}, L^{c}) → ∞, as k → ∞.  (3.20)

Also, for any sequence z_{k} ∈ U^{c}, the estimate

ρ′_{k}(z_{k}) ≥ ρ′_{k}(y_{k})\frac{dist_{γ′_{k}}(z_{k}, L^{c})}{dist_{γ′_{k}}(y_{k}, L^{c})},  (3.21)

follows from the fact that the ratio (3.18) is scale invariant and is minimized at y_{k}. Thus, the sequence (U^{c}, γ′_{k}, y_{k}) has L^{2} curvature radius uniformly bounded below, at points within a bounded but arbitrary distance from y_{k}.

Next, recall that the volume radius (1.15) scales as a distance. By (3.9), (3.21) and scale invariance, we have a uniform lower bound ν′_{k}(z_{k}) ≥ ν_{o} ·ρ′(z_{k}) ≥ ν_{o} ·C(dist(z_{k}, y_{k})). Thus the sequence (U^{c}, γ′_{k}, y_{k}) cannot collapse anywhere. From Theorem 1.5, it follows that this pointed sequence has a subsequence which converges uniformly on compact subsets, in the weak L^{2,2} topology, to a limit L^{2,2} Riemannian manifold (N, γ′, y). The estimate (3.20) further implies that the metric γ′ is complete; the distance to the boundary of U^{c}, namely L^{c}, goes to infinity as k → ∞.

From the arguments in Proposition 3.1, c.f. (3.10)-(3.13), we see that

(L′)^{∗}u_{k} → 0, strongly inL^{2}(γ′_{k}).  (3.22)

Now by definition, u_{k} ≥ c in U^{c}. If u_{k}(y_{k}) → ∞, consider the function

ū_{k}(z_{k}) = \frac{u_{k}(z_{k})}{u_{k}(y_{k})},  (3.23)

so that ū_{k} ≥ 0, ū_{k}(y_{k}) = 1. Otherwise, let ū_{k} = u_{k}. The trace equation associated to (3.22) reads

Δ′ū_{k} + s′_{k}ū_{k} = (u_{k}(y_{k}))^{−1}(tr ξ′_{k} + s′_{k}).  (3.24)

Note that the right-hand side of (3.24) goes to 0 in L^{2}. It follows from the lower bound ū_{k} ≥ 0 and the DeGiorgi-Nash-Moser estimates, c.f. [GT, Thms 8.17, 8.18, 8.22], that the oscillation of ū_{k} is uniformly bounded in B′_{y_{k}}(\frac{1}{2}). Applying the same considerations to neighboring balls of radius \frac{1}{2}ρ′, it follows that the oscillation of ū_{k} is uniformly bounded in B′_{z_{k}}(\frac{1}{2}ρ′(z_{k})) for all z_{k} within uniformly bounded distance to y_{k}. This, together with standard L^{2} estimates for elliptic equations, c.f. [GT, Thm.8.8] imply that a subsequence of {ū_{k}} converges, in the weak L^{2,2} topology, with respect to the metrics γ′_{i}, to a limit L^{2,2} function ū on N. Further, by (3.22) and the fact that (3.22) is preserved under the renormalization (3.23), the limit pair (γ′, ū) satisfies

L′^{∗}ū = 0, Δ′ū = 0,  (3.25)

so that (N, γ′, ū) is a weak solution of the static vacuum Einstein equations.

Note that the (weakly) harmonic function ū satisfies ū ≥ 0 everywhere and ū(y) = 1. By the (weak) maximum principle, c.f. [GT, Thm.8.1], in fact ū > 0 everywhere. As noted in §1.3, when the potential ū > 0, elliptic regularity implies that L^{2,2} weak solutions of (3.25) are C^{∞}. From Theorem 3.2(I), it follows that γ′ is flat, and ū is constant.

On the other hand, we have the estimate (3.19) for the sequence (γ′_{k}, y_{k}). The fact that the convergence of γ′_{k} to γ′ is only in the weak L^{2,2} topology does not imply that (3.19) passes continuously to the limit. However, we will show in Theorem 3.4 below that in fact the convergence γ′_{k} → γ′ is in the strong L^{2,2} topology. By (1.28), the radius ρ(x) is continuous in the strong L^{2,2} topology and one thus obtains the estimate

ρ′(y) = 1,

on the limit. This however contradicts the fact that γ′ is flat; a complete flat manifold (N, g) clearly has ρ(x) = ∞, at any x.

Hence the proof of Theorem 3.3 is completed with the proof of the following result, which will also be used frequently in the work to follow.

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Theorem 3.4 (Strong Convergence). Let {g_{i}} be a sequence of Yamabe metrics, (not necessarily of volume 1), with associated splittings

L^{∗}u_{i} + ξ_{i} = −\frac{s_{i}}{3} · g_{i}.  (3.26)

Suppose that for all i, there is a constant d > 0 such that

r_{h}(x_{i}, g_{i}) ≥ 1, and r_{h}(y_{i}, g_{i}) ≥ d, ∀y_{i} ∈ ∂B_{x_{i}}(1),  (3.27)

where r_{h} denotes the L^{2,2} harmonic radius. Suppose also that ξ_{i} → 0 strongly in L^{2}(B_{x_{i}}(1)), s_{i} ≥ −s_{o}, for some s_{o}, and that there is a constant u_{o} > 0 such that

u_{i} ≥ u_{o} on B_{x_{i}}((1 + \frac{1}{2}d)).  (3.28)

Then on B_{x_{i}}(1), a subsequence of {g_{i}} converges strongly in the L^{2,2} topology to a limit metric g_{o} on B_{x}(1), where x = lim x_{i}.

Proof. Let B = B_{i} = B_{x_{i}}(1) and set B′ = B_{x_{i}}(1 + \frac{d}{2}). By (3.27) and Theorem 1.5, we may assume that a subsequence of {g_{i}} converges weakly in the L^{2,2} topology to a limit metric g_{o} on B′. Thus, there are coordinates {y_{k}} on suitable balls D ⊂ B_{x_{i}}(1 + d), covering B′, such that the functions

(g_{i})_{kl} → (g_{o})_{kl},  (3.29)

weakly in L^{2,2}(D). As in the proof of Theorem 3.3, (by dividing by u(x_{i}) if necessary, see the discussion following (3.23)), we may assume that the functions u_{i} are uniformly bounded, in L^{∞}(B′), and that (3.28) holds, possibly with a different constant u_{o}. In particular, a subsequence converges strongly in L^{2} to a limit function u. Thus, from the trace equation

Δ_{i}u_{i} = tr ξ_{i} − s_{i}u_{i} + s_{i},  (3.30)

and the arguments above, together with the assumption on ξ, Δ_{i}u_{i} converges strongly in L^{2} to a limit L^{2} function Δ_{o}u on B′ with u ∈ T^{2,2}(B′).

We now use the L^{2} estimates for elliptic equations of the form (3.30), see [GT, Thm 8.8]. This gives

||D^{2}u_{i}||_{L^{2}(B)} ≤ C(||Δ_{i}u_{i}||_{L^{2}(B′)} + ||u_{i}||_{L^{2}(B′)}),  (3.31)

where C depends on the L^{2,2} norm of the metrics g_{i} and the constant d in (3.27). Here the norms and derivatives D^{2} are taken with respect to the fixed coordinates {y_{k}} and limit metric g_{o}. From this, it then follows first that

||D^{2}u_{i}||_{L^{2}(B)} ≤ C,  (3.32)

so that a subsequence of {u_{i}} converges weakly in the L^{2,2} topology to u. By the Sobolev embedding theorem, {u_{i}} (sub)-converges strongly in the L^{1,2} topology. Repeating the estimate (3.31) on u − u_{i} gives

||D^{2}(u − u_{i})||_{L^{2}(B)} ≤ C(||Δ_{i}(u − u_{i})||_{L^{2}(B′)} + ||(u − u_{i})||_{L^{2}(B′)}).  (3.33)

Clearly ||(u − u_{i})||_{L^{2}(B′)} → 0. Write Δ_{i}(u − u_{i}) = (Δ_{i}u − Δ_{o}u) + (Δ_{o}u − Δ_{i}u_{i}). From preceding arguments, Δ_{i}u_{i} → Δ_{o}u strongly in L^{2}(B′). Consider then the sequence Δ_{i}u − Δ_{o}u. We have in the y_{k}-coordinates

Δ_{i}u = g_{i}^{kl}∂_{kl}u + ∂g_{i}^{kl}∂_{k}u.  (3.34)

Since {g_{i}} is uniformly bounded in L^{2,2}, a subsequence converges strongly in L^{1,2} ∩ L^{∞} to g_{o}. This shows that Δ_{i}u converges strongly in L^{2} to Δ_{o}u. Thus, (3.33) implies that

||D^{2}(u − u_{i})||_{L^{2}(B)} → 0.  (3.35)

If D_{i}^{2} denotes the Hessian with respect to the metrics g_{i}, the same reasoning as above on the Laplacian in (3.34) then also gives

D_{i}^{2}u_{i} → D^{2}u, strongly in L^{2}(B).  (3.36)

Now return to the splitting equation (3.26), which we write as

u_{i}r_{i} = D_{i}^{2}u_{i} − Δ_{i}u_{i} + ξ_{i} + \frac{s_{i}}{3} · g_{i}.  (3.37)

By the preceding arguments, the right hand side of (3.37) converges strongly in L^{2} to its limit on B and thus

∫_{B} |u · r_{o} − u_{i} · r_{i}|^{2}dV_{o} → 0,  (3.38)

where the norm is taken in the g_{o} metric and r_{o} is the weak L^{2} limit of the Ricci curvature r_{i} of g_{i}. Since u_{i} → u in L^{2,2} ∩ C^{o} on B and since u_{i} ≥ u_{o} by (3.28), it follows that

∫_{B} |r_{o} − r_{i}|^{2}dV_{o} → 0,  (3.39)

so that the Ricci curvature of {g_{i}} converges strongly in L^{2} to the Ricci curvature of g_{o} on B. The same reasoning shows that r_{i} converges strongly to r on a thickening, say B(1 + \frac{d}{4}) of B. This implies the strong L^{2,2} convergence of g_{i} to g_{o}, via the equation for the Ricci curvature in harmonic coordinates on B, see [An3, p.434] for the details here.

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Remark 3.5. (i): If one drops the assumption (3.27) on the behavior of r_{h} at the boundary, so that the sequence (B_{x_{i}}(1), g_{i}) satisfies r_{h}(x_{i}, g_{i}) = 1 with u_{i} ≥ u_{o} on B_{x_{i}}(1) and ξ_{i} → 0, s_{i} ≥ −s_{o}, then Theorem 3.4 implies that a subsequence converges strongly in the L^{2,2} topology on B_{x_{i}}(1 − δ), for any fixed δ > 0, to a limit metric g_{o} on B_{x}(1 − δ).

(ii): Returning to Theorem 3.3, note that the estimate (3.17), together with Lemma 1.4, implies that if u_{i} ≥ u_{o} > 0 on M, then

∫_{M} |z_{i}|^{2}dV_{i} ≤ C/u_{o}.  (3.40)

This has already been proved by elementary means in (2.18).

(iii): We note that exactly the same proof as Theorem 3.3 shows the estimate (3.17) holds in the region U_{−c} = {x ∈ M : u < −c < 0}, i.e. ρ(x) ≥ a · dist(x, L^{−c}), for x ∈ U_{−c}.

§3.4. The discussion in §3.3 implies that ρ(x_{i}) can become arbitrarily small, under the sequence {g_{i}}, only if the 0-level set L^{o}, or an ε-level set L^{ε} for ε very small, comes arbitrarily close to x_{i}.

In this subsection, we will begin the study of the geometry of the degeneration of (M, g_{i}) in a neighborhood of the 0-level L^{o}. On the other hand, from the remarks above, the 0-level itself must be avoided; see also Remark 3.15(i).

The treatment of the three possible cases σ(M) < 0, σ(M) = 0 and σ(M) > 0 is somewhat different. Some indication for the need to treat these cases separately is already apparent from §2. Thus, throughout §3.4, we assume

σ(M) < 0,

or more precisely that {g_{i}} is a sequence of unit volume Yamabe metrics on M with scalar curvature bounded away from 0 and −∞, i.e.

−∞ < −s_{o} ≤ s_{g_{i}} ≤ −s_{1} < 0.  (3.41)

The cases σ(M) = 0 and σ(M) > 0 are treated in §3.5 and §3.6 respectively. As in Theorem A(II), c.f. (0.19), we also assume

∫_{M} |z^{T} |^{2}dV ≤ K,  (3.42)

and

ν(x) ≥ ν_{o} · ρ(x);  (3.43)

note that (3.43) is more general than (0.18).

To set the stage for the considerations to follow, let T_{i} = max u_{i} and choose x_{i} such that

|u_{i}(x_{i})/T_{i} − 1| → 0, as i → ∞.  (3.44)

Thus, we are considering points as far away as possible, in terms of the function u, from the 0-levels of u. By (2.35), as discussed in Remark 2.12, (3.41) and (3.42) imply one has a uniform lower bound

T_{i} ≥ λ_{o} ≥ 0.  (3.45)

Note: The bound (3.45) is the only place in §3.4, (and also in §3.6), where the assumption (3.42) is used. In §3.5, corresponding to the case σ(M) = 0, (3.42) is used in a stronger way.

Consider ρ(x_{i}), the L^{2} curvature radius at x_{i}, of course with respect to the metric g_{i}. If there exists a uniform lower bound

ρ(x_{i}) ≥ ρ_{o}, ∀i,  (3.46)

then Theorem 1.5 provides a complete description of the possible behavior of {g_{i}} in B_{x_{i}}(ρ_{o}). Suppose instead, for a possibly different choice {x_{i}^{1}}, that both

\begin{matrix}|u_{i}(x_{i}^{1})/T_{i} − 1| → 0, and \\ ρ(x_{i}^{1}) → 0, as i → ∞. \end{matrix}  (3.47)

Thus, the metrics g_{i} are degenerating in a (progressively smaller) neighborhood of x_{i}^{1} and this degeneration cannot be described by Theorem 1.5.

To understand the degeneration of (M, g_{i}) at or near {x_{i}^{1}}, rescale the metrics g_{i} by ρ(x_{i}^{1})^{−2}. Thus, set

g_{i}^{1} = ρ(x_{i}^{1})^{−2} · g_{i},  (3.48)

and let ρ^{1} denote the L^{2} curvature radius with respect to the rescaled metrics g_{i}^{1}. One then has

ρ^{1}(x_{i}^{1}) = 1, for all i,  (3.49)

so that Theorem 1.5 and (3.43) imply the sub-convergence of {g_{i}^{1}} in B_{x_{i}^{1}}^{1}(1). Thus we may assume that {g_{i}^{1}} converges, weakly in the L^{2,2} topology, to a limit L^{2,2} metric g^{1}, defined on B^{1} = B_{x^{1}}^{1}(1), with base point x^{1} = lim x_{i}^{1}.

Proposition 3.6. The limit (B^{1}, g^{1}, x^{1}) is a flat solution of the static vacuum equations (3.1), with potential function

ū = lim(u_{i}/T_{i}) ≡ 1 on B^{1}.  (3.50)

Proof. The splitting (2.10) of g_{i}, when rescaled, gives the splitting for g_{i}^{1}. Thus

\begin{matrix}(L^{∗})^{1}(u_{i}) + ξ_{i}^{1} = −\frac{s_{i}^{1}}{3} · g_{i}^{1}, \\ Δ^{1}u_{i} + s_{i}^{1}u_{i} = tr ξ_{i}^{1} + s_{i}^{1}. \end{matrix}  (3.51)

As in the proof of Proposition 3.1, we have ξ_{i}^{1} → 0 in L^{2}(M, g_{i}^{1}), s_{i}^{1} → 0 in L^{∞}(M, g_{i}^{1}). While it may be possible apriori that u_{i}(x_{i}^{1}) → +∞, (corresponding to the possibility that T_{i} → ∞), as in the proof of Theorem 3.3, (or Proposition 3.1) we renormalize equation (3.51) by dividing T_{i} to obtain the (renormalized) limit function ū on B^{1}. As noted above, T_{i} is bounded away from 0 by (3.45), (a consequence of the standing assumption (3.42)), so that the renormalizations of the terms ξ_{i}^{1}, s_{i}^{1} go to 0 at least as fast as before.

Thus, on B^{1}, the limit metric g^{1} satisfies the static vacuum Einstein equations

\begin{matrix}(L^{∗})^{1}(ū) = 0, \\ Δ^{1}ū = 0, \end{matrix}  (3.52)

where ū = lim u_{i}/T_{i}. As in the proof of Theorem 3.3, the convergence of ū_{i} = u_{i}/T_{i} to the limit is in the weak L^{2,2} topology, and since ū_{i}(x_{i}^{1}) = 1, the limit ū is a weakly harmonic L^{2,2} function, which is not identically 0.

As discussed in §1.3, the equations (3.52) imply in particular that the metric g^{1} is smooth away from the 0-locus of ū. Now by (3.44), x^{1}, the center of the ball B^{1}, satisfies

1 = ū(x^{1}) ≥ ū(y),  (3.53)

for all y ∈ B^{1}. Since ū is harmonic, it follows that ū ≡ 1 and thus (B^{1}, g^{1}) is flat.

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The convergence g_{i}^{1} → g^{1} is in the strong L^{2,2} topology, on B_{x_{i}^{1}}^{1}(s), for any fixed s < 1, by Theorem 3.4. It thus follows that all of the curvature of {g_{i}^{1}} in L^{2} is concentrating on the boundary ∂B_{x_{i}^{1}}^{1}(1). This implies that the limit (B^{1}, g^{1}) itself does not yet effectively model the degeneration of g_{i} near x_{i}^{1}; the base points x_{i}^{1} must be altered slightly.

The fact that some or all of the curvature in L^{2} is concentrating on ∂B_{x_{i}^{1}}^{1} implies, (and is equivalent to),

ρ^{1}(y_{i}) → 0,

for some sequence y_{i} ∈ ∂B_{x_{i}^{1}}^{1}(1), or, what is same, ρ_{i}(y_{i}) << ρ_{i}(x_{i}^{1}). Thus the curvature is blowing up at y_{i} much faster than at x_{i}^{1}. This is the first indication that the curvature of (M, g_{i}) blows up at many different scales.

The preceding remarks lead naturally to the following definitions.

Definition 3.7. A point y in a complete Riemannian manifold (N, γ) is (ρ, c) buffered if c > 0 and

\frac{ρ(y)^{4}}{vol B_{y}(ρ(y))} ∫_{B_{y}((1−c)·ρ(y))} |r|^{2} ≥ c · c_{o},  (3.54)

where c_{o} is the constant in the definition of ρ, c.f. (1.26).

Similarly, y is strongly (ρ, d) buffered if d > 0 and

ρ(z) ≥ d · ρ(y),  (3.55)

for all z ∈ ∂B_{y}(ρ(y)).

A sequence of points y_{i} in the manifolds (N, γ_{i}) is (ρ, c) buffered, or strongly (ρ, c) buffered, if each y_{i} is.

The buffer constants c and d are arbitrary small parameters, as is the parameter c_{o}; their precise values, beyond being small, are not important. From §1, c_{o} is a fixed small number throughout the paper, e.g. c_{o} = 10^{−3}. The buffer constant c, (more important than d), might be allowed to vary over small numbers, say 0 < c < 10^{−3}, but in any given discussion, the value of c will be fixed.

Note that the strong (ρ, d) buffered condition appears in Theorem 3.4 (strong convergence), and is invariant under scaling. Thus, if {g_{i}} is a sequence of unit volume Yamabe metrics on a closed 3-manifold M with ρ(y_{i}) → 0, then the blow up metrics g′_{i} = ρ(y_{i})^{−2} · g_{i} converge strongly in L^{2,2} on B′_{y_{i}}(1), to a non-flat limit, provided the sequence {y_{i}} is strongly (ρ, d) buffered, and u_{i} is bounded away from 0 on B′_{y_{i}}(1). Thus, this condition prevents any of the curvature from concentrating in L^{2} (with normalized measure) on the boundary.

Similarly, the (ρ, c) buffered condition (3.54) is invariant under scaling and prevents all of the curvature from concentrating in L^{2} on the boundary. Thus, if c is close to 0, then a definite (small) percentage of the curvature in L^{2} is in the ball of radius (1 − c). (The situation where c is close to 1 will never arise here). In particular, from Theorem 3.4, any limit B′_{y}(1) of a (ρ, c) buffered sequence (B′_{y_{i}}(1), g′_{i}, y_{i}) of Yamabe metrics as above, with u_{i} bounded away from 0 in B′_{y_{i}}(1), cannot be flat, since one has strong convergence everywhere in the interior. Conversely, again by Theorem 3.4, if u_{i} is bounded away from 0 on B′_{y_{i}}(1) and the limit is not flat, then the sequence is (ρ, c) buffered, for some c > 0, c.f. also Remark 3.5(i).

Note that from the definition, it is obvious that if a given sequence is (ρ, c) buffered, then it is (ρ, c′) buffered, for any c′ < c, and similarly for strongly buffered sequences.

The next Lemma formalizes the relation between these notions, for Yamabe metrics.

Lemma 3.8. Let g be a unit volume Yamabe metric on a closed 3-manifold M, and y a base point in M which is strongly (ρ, d) buffered. Suppose u(y) = 1, the non-collapse assumption (3.43) holds at y, and the blow-up (B′_{y}(1), g′, y), g′ = ρ(y)^{−2} · g is ϵ-close to a static vacuum solution, in the sense that ||L^{∗}u||_{L^{2}} ≤ ϵ in B′_{y}(1), as in (3.13).

Under these assumptions, there exists ε_{o} > 0 such that if ε ≤ ε_{o}, and if either

(i) u_{i} is bounded away from 0 in B′_{yi}(1), or

(ii) |u_{i} − 1| ≤ \frac{1}{4} in B′_{yi}(\frac{1}{2}), then there is a constant c = c(d, c_{o}, ν_{o}) > 0 such that y is (ρ, c) buffered.

Proof. The proof is by contradiction. Thus, if the conclusion is not true, there exists a pointed sequence (M, g_{i}, y_{i}) satisfying the assumptions with ε = ε_{i} → 0, such that the sequence y_{i} is not (ρ, c) buffered. Let (B′(1), g′, y) be a weak L^{2,2} limit of (a subsequence of) (B′_{y_{i}}(1), g′_{i}, y_{i}), so that (B′(1), g′) is an L^{2,2} static vacuum solution. The existence of such limits follows, as previously in §3.4, from the non-collapse assumption (3.43).

By the discussion above regarding strong convergence, it suffices to prove that u_{i} is bounded away from 0 on B′_{y_{i}}(1) and the limit is not flat. In the case of assumption (i), this first condition is obvious, while the second follows since ρ is continuous under strong L^{2,2} convergence. Hence one has a contradiction in this case.

With regard to (ii), suppose first the limit (B′(1), g′) were flat. Then from the static vacuum equations, the limit function u is an affine function on B′(1). The assumption (ii) then implies that u is uniformly bounded away from 0 in B′(1), in fact u ≥ \frac{1}{2}. Further, since y_{i} is strongly (ρ, d) buffered, the functions u_{i} are uniformly controlled in L^{2,2}, and hence controlled in C^{α}, α < \frac{1}{2}, in B′_{y_{i}}(1 + \frac{d}{2}). It follows that u_{i} is uniformly bounded away from 0 in B′_{y_{i}}(1). As in Case (i) above, this implies that the full limit B′(1) could not have been flat.

Since the full limit (B′(1), g′) is not flat, it follows from the real-analyticity of static vacuum solutions discussed in §1.3, and (ii) again, that the limit is not flat in any domain in B′(1) on which u is bounded away from 0; in particular, this is the case on B′(\frac{1}{2}). Hence, as above, u_{i} is bounded away from 0 in B′_{y_{i}}(\frac{1}{2}), and the strong convergence to this limit B′(\frac{1}{2}) implies that the sequence is (ρ, c) buffered, for some c > 0.

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The buffer constant c = c(d, c_{o}, ν_{o}) in Lemma 3.8 could be explicitly estimated in terms of d, c_{o} and ν_{o}; however, we will have no need to do this. Of course, c may well be much smaller than d. In the following, we will usually use the contrapositive of Lemma 3.8, namely that if a given sequence y_{i} is not (ρ, c) buffered, for a given c > 0, then it is not strongly (ρ, c) buffered if either (i) or (ii) in Lemma 3.8 hold; here d = d(c, c_{o}, ν_{o}) may be large compared with c, but may made arbitrarily small by choosing c sufficiently small.

Now we return to the situation preceding Proposition 3.6. Of course the points x_{i}^{1} in (3.47) above are not (ρ, c) buffered, for any choice of c > 0. Observe that all the positive level sets L^{s}, s > 0, of ū_{i} have points converging to ∂B_{x_{i}^{1}}^{1}(1). For otherwise, by Theorem 3.3, {x_{i}^{1}} would be a (strongly) buffered sequence, a contradiction.

Let U = U(i) be the component of (u_{i}/T_{i})^{−1}((0, 2]) ⊂ M containing x_{i}^{1}. Let L^{k} = L^{k}(i), k ≥ 1, denote the level set (u_{i}/T_{i})^{−1}(2^{−k+1}) in U(i), (possibly having many components). By a slight perturbation of the values, we may assume that L^{k} are smooth hypersurfaces. Let U^{k} = U^{k}(i) be the set u_{i}^{−1}[2^{−k+1}, 2] ⊂ U. The choice of the factor 2 here is for convenience, and may be replaced by any fixed number > 1.

In Theorem 3.10 below, we will construct a (ρ, c)-buffered sequence from an initial sequence {x_{i}^{1}} satisfying conditions similar to (3.47) above, in order to produce a non-flat blow up limit. This is done essentially by ‘descending down’ the level sets L^{k} of u_{i}, for i fixed.

To understand the underlying idea and motivation of the proof of Theorem 3.10, it is useful to point out that the descent down the levels of u must require an infinite process. This is shown by the following remark and example. Remark 3.9 and Example 3.9 below are not logically necessary in this respect, and may be skipped if preferred.

Remark 3.9. Recall again that the metrics g_{i}^{1} converge uniformly on compact subsets of B_{x_{i}^{1}}^{1}(1) to the flat metric, and u_{i}/T_{i} → 1 uniformly on such compact subsets. It is clear that there exists a sequence {x_{i}^{2}} ∈ U^{2} ∩ B_{x_{i}^{1}}^{1}(1) such that

ρ^{1}(x_{i}^{2}) → 0, i.e. ρ(x_{i}^{2}) << ρ(x_{i}^{1}),  (3.56)

and consequently

dist_{g_{i}^{1}}(x_{i}^{2}, ∂B_{x_{i}}^{1}(1)) → 0.

There are many possible choices of {x_{i}^{2}} satisfying (3.56).

For reasons that will be apparent below, we require further that {x_{i}^{2}} satisfy the following: r_{i}^{2} ≡ dist_{g_{i}^{1}}(x_{i}^{2}, x_{i}^{1}) → 1 as i → ∞, and

|u_{i}(x) − u_{i}(x_{i}^{1})| ≤ μ_{o} · u_{i}(x_{i}^{1}), ∀x ∈ B_{x_{i}^{1}}^{1}(r_{i}^{2}).  (3.57)

Here μ_{o} is a fixed constant, 0 < μ_{o} < \frac{1}{2}, for example μ_{o} = \frac{1}{4}. Again it is clear that there are many possible choices of such points x_{i}^{2}. If T_{i} = max u_{i} → ∞, (3.57) is assumed to apply to the renormalized functions u_{i}/T_{i}, as before. The factor T_{i} will be ignored below.

Now rescale the metrics g_{i}^{1} further to make ρ(x_{i}^{2}) = 1, i.e. set

g_{i}^{2} = ρ^{1}(x_{i}^{2})^{−2} · g_{i}^{1} = ρ(x_{i}^{2})^{−2} · g_{i}.

Consider then the sequence of pointed Riemannian manifolds (B_{x_{i}^{2}}^{2}(1), g_{i}^{2}, x_{i}^{2}). This has L^{2} curvature radius equal to 1 at x_{i}^{2}, and as above, one may apply Theorem 1.5. The collapse case is ruled out by assumption (3.43).

Thus a subsequence of {g_{i}^{2}} converges, in the weak L^{2,2} topology on B_{x_{i}^{2}}^{2}(1), and in the strong L^{2,2} topology on B_{x_{i}^{2}}^{2}(s), any s < 1 away from the locus {|u_{i}| ≤ ε} for any given ε > 0, to a limit (B^{2}, g^{2}, x^{2}). Necessarily, this limit is a solution of the static vacuum Einstein equations (3.1), with limit function u^{2}. Note that since u_{i}(x_{i}^{2}) ≥ 1−2μ_{o} for i large, the limit function u^{2} is not identically 0.

However, we claim that the limit (B^{2}, g^{2}) is also flat and u^{2} is constant. To see this, consider first the ‘half-ball’ B_{x_{i}^{2}}^{2}(1) ∩ B_{x_{i}^{1}}^{2}(R_{i}^{2}), where R_{i}^{2} = dist_{g_{i}^{2}}(x_{i}^{2}, x_{i}^{1}). Note that in this scale, R_{i}^{2} → ∞ as i → ∞ by (3.56)-(3.57). Since the curvature of B_{x_{i}^{1}}^{1}(1) is bounded, and g_{i}^{2} is a ’blow-up’ of g_{i}^{1}, the curvature of (D_{i}^{2}, g_{i}^{2}) goes to 0 uniformly in L^{2}. Hence the limit half-ball (D^{2}, g^{2}) is flat.

To extend this to the full ball B^{2}, the static vacuum equations imply that D^{2}u^{2} = 0, so that u^{2} must be an affine function on its domain. We claim that the limit harmonic function u^{2} on D^{2} is constant, i.e.

u^{2} = const. ≥ 1 − μ_{o}.  (3.58)

To see this, return to the balls B_{x_{i}^{2}}^{2}(1) in the scale g_{i}^{2}. In this scale, the previous ball (B_{x_{i}^{1}}^{1}(1), g_{i}^{1}) is expanded into a ball B_{x_{i}^{1}}^{2}(R_{i}) of radius R_{i} → ∞ as i → ∞. Let γ_{i}^{2} be a minimizing geodesic from x_{i}^{2} to x_{i}^{1} in B_{x_{i}^{1}}^{2}(R_{i}), parametrized by arclength w.r.t. g_{i}^{2}. Note that the length of γ_{i}^{2} becomes arbitrarily large. Now suppose that u_{i}^{2} converges to a non-constant affine function u^{2} on the limit B^{2}. Note that all the (horo)-balls B_{γ_{i}(t)}^{2}(t), centered at γ_{i}(t) and containing x_{i}^{2} at the boundary, also converge to flat balls in the limit, for any given t. (Here we recall from the definition of ρ, c.f.(1.27), that ρ^{2}(x) ≥ dist_{g_{i}^{2}}(x, ∂B_{x_{i}^{1}}^{2}(R_{i})). It follows that the functions u_{i} approximate the unique non-constant affine function u^{2} in B_{γ(t)}^{2}(t) extending the limit function u^{2} in B^{2}. From the assumption (3.57) on the choice of x_{i}^{2}, it follows that |u_{i} − 1| ≤ 2μ_{o} everywhere in B_{γ_{i}(t)}^{2}(t) and hence the same holds on the limit. But clearly a non-constant affine function cannot have this property. This proves second claim above.

Now (B^{2}, g^{2}, u^{2}) is an L^{2,2} static vacuum solution, and from the above, (D^{2}, g^{2}) is flat, with u^{2} a positive constant. Since, from the discussion in §1.3, L^{2,2} static vacuum solutions are real-analytic away from the 0-locus of the potential, it follows that (B^{2}, g^{2}) itself is flat, and u^{2} is constant in B^{2}, as claimed.

Thus the geometry of the sequence (B_{x_{i}^{2}}^{2}(1), g_{i}^{2}, x_{i}^{2}) and that of the potential function u_{i} on B_{x_{i}^{2}}^{2}(1) is exactly the same as on the previous sequence (B_{x_{i}^{1}}^{1}(1), g_{i}^{1}, x_{i}^{1}). Clearly one may repeat this process an arbitrary finite number of times; just return to (3.56)-(3.58) and raise the indices successively by 1. However, each limit g^{k} constructed in this way is flat with constant potential function and hence the sequence {x_{i}^{k}} is not (ρ, c) buffered, for any fixed c > 0. Thus, no finite iteration of this process leads to a (ρ, c) buffered sequence.

It is worthwhile to examine the construction in Remark 3.9 on a specific example.

Example 3.9 Let (N, g) be the Schwarzschild metric (0.17), the canonical solution of the static vacuum equations. Let x_{j}^{1} be a divergent sequence of points in N, so that t(x_{j}^{1}) → ∞, where t is the distance to the event horizon. Note that the curvature z of (N, g) decays on the order of t^{−3} as t → ∞ and u(x_{j}^{1}) → 1 = max u.

Now blow-down the metric g based at {x_{j}^{1}}, i.e. form the ‘tangent cone at infinity’ based at {x_{j}^{1}}, by considering the metrics g_{j}^{1} = ρ(x_{j}^{1})^{−2} · g. The metrics g_{j}^{1} converge to the flat metric on B_{x^{1}}^{1}(1), x^{1} = lim x_{j}^{1} and the functions u_{j}^{1} given by the restriction of u to B_{x_{j}^{1}}^{1}(1) converge to the constant function 1.

Of course the sequence (B_{x_{j}^{1}}^{1}(1), g_{j}^{1}, x_{j}^{1}) is not (ρ, c) buffered, for any c > 0. A short computation, using the cubic curvature decay and the definition of ρ, shows that

ρ(x_{j}^{1}) ∼ t(x_{j}^{1}) − (t(x_{j}^{1}))^{1/3}.

Thus, as in the construction in Remark 3.9, consider a second sequence of points x_{j}^{2}, with t(x_{j}^{2}) ∼ (t(x_{j}^{1}))^{1/3}. Again if one blows-down the metric g based at {x_{j}^{2}}, then the metrics g_{j}^{2} = ρ(x_{j}^{2})^{−2} · g converge to the flat metric and the functions u_{j}^{2}, now given by the restriction of u to B_{x_{j}^{2}}^{2}(1) again converge to the constant function 1.

This process may be repeated any finite number of times, so that by induction, points x_{j}^{k} are chosen satisfying

t(x_{j}^{k}) ∼ (t(x_{j}^{1}))^{1/3^{k}}.

However one always obtains flat limits in this manner. Observe that for any given choice of k, the functions u_{j}^{k} defined as above, always converge to the constant function 1. In fact u(x_{j}^{k}) ∼ 1 − t^{−1}(x_{j}^{k}) → 1 as j → ∞, for any fixed k. It is clear that to obtain a non-flat limit, (namely the original Schwarzschild metric), one must increase the descent level k in {x_{j}^{k}} as a function of j, with k = k(j) → ∞ as j → ∞. One obtains a non-flat limit when the base points x_{i}^{k} satisfy u(x_{i}^{k}) = 1 − δ, for any fixed δ > 0. In this particular example, even though k(j) → ∞ as j → ∞, the potential function u_{j}^{k(j)} does not converge to the 0-function as j → ∞.

We now return to the main issue of constructing a (ρ, c) buffered sequence from some initial sequence whose blow-up limits are flat, with constant potential function.

Theorem 3.10 (Existence of Non-Flat Blow-ups). Let {g_{i}} be a sequence of unit volume Yamabe metrics on a closed 3-manifold M, satisfying the non-collapse assumption (3.43) and scalar curvature bound (3.41), i.e.

−∞ < −s_{o} ≤ s_{g_{i}} ≤ −s_{1} < 0.  (3.59)

Suppose there are points x_{i} ∈ (M, g_{i}) and a number δ_{o} > 0 such that

ρ(x_{i}) → 0, as i → ∞, and u_{i}(x_{i})/T_{i} ≥ δ_{o} > 0,  (3.60)

where T_{i} ≡ max_{M} u_{i} satisfies (3.45), i.e. for some λ_{o} > 0,

T_{i} ≥ λ_{o}.  (3.61)

Then for any fixed c > 0 sufficiently small, there exists a sequence of (ρ, c)-buffered base points {y_{i}}, satisfying the following conclusions: u_{i}(y_{i}) ∈ (0, T_{i}), dist_{g_{i}}(x_{i}, y_{i}) → 0, and the rescalings

g′_{i} = ρ_{i}^{−2}(y_{i}) · g_{i}, ρ_{i}(y_{i}) → 0,  (3.62)

of {g_{i}} sub-converge, in the weak L^{2,2} topology based at {y_{i}} on a domain D, to a limit smooth metric g′ on D, based at y′ = lim y_{i}. The pair (D, g′) is a non-flat, (and non-super-trivial), solution to the static vacuum Einstein equations, with limit potential function ū constructed from {u_{i}}. The limit domain D is naturally embedded in M via the metric g′ and the convergence is in the strong L^{2,2} topology away from the locus {ū = 0}.

Proof. The proof proceeds in several steps, organized around several lemmas. Recall again that the estimate (3.61) follows from a bound on the L^{2} norm of z^{T} under the bounds (3.59), c.f. (3.45).

First, as previously in Proposition 3.1 and Remark 3.9 for example, it is necessary to renormalize u_{i} to ū_{i} = u_{i}/T_{i}, at least if T_{i} → ∞. We will assume this is done, so that sup ū_{i} = 1, and neglect the bar notation on ū_{i}.

Step I. To begin, the sequence {x_{i}} satisfying (3.60) must be altered to a new sequence of points {x_{i}^{1}}, still satisfying (3.60), but with (locally) near-maximal u-values, (compare with (3.47)).

Lemma 3.11. (Initial Degeneration Points). Under the assumption (3.60), there exists a subsequence {i′} of {i}, relabeled to {i}, and base points x_{i}^{1}, with dist_{g_{i}}(x_{i}, x_{i}^{1}) → 0 which still satisfy (3.60), such that for any point q_{i} ∈ B_{x_{i}^{1}}(ρ(x_{i}^{1})),

u_{i}(q_{i}) ≤ u_{i}(x_{i}^{1}) + ε_{i},  (3.63)

for some sequence ε_{i} = ε(x_{i}^{1}) → 0 as i → ∞.

Proof. The proof is based on the fact that ρ is a Lipschitz function with Lipschitz constant 1, c.f. (1.27). Thus given a sequence x_{i} satisfying (3.60), suppose (3.63) does not hold in B_{x_{i}}(ρ(x_{i})). Then there exist q_{i} ∈ B_{x_{i}}(ρ(x_{i})) such that u(q_{i}) ≥ u(x_{i}) + ε_{o}, for some ε_{o} > 0. But ρ(q_{i}) ≤ 2ρ(x_{i}) → 0 as i → ∞, so that {q_{i}} also satisfies (3.60). Clearly this procedure may be repeated any given number of times. Since max u_{i} = 1, it follows that for any given ε_{o} > 0, there exist x_{i}^{ε_{o}} such that, for all i sufficiently large, (3.63) holds with ε_{o} in place of ε_{i} and ρ(x_{i}^{ε_{o}}) → 0 as i → ∞. Now replace ε_{o} by a sequence ε_{j} → 0 and choose a suitable diagonal subsequence j_{i} of {i, j} with ε_{j_{i}} → 0 sufficiently slowly as i → ∞. Then the base points x_{i}^{1} ≡ x_{i}^{ε_{j_{i}}} satisfy (3.60) and (3.63).

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From now on, we work with the subsequence {x_{i}^{1}} from Lemma 3.11. Of course the choice of {x_{i}^{1}} may be far from unique. Even more, sequences satisfying (3.60) may also be far from unique. The initial points x_{i}^{1} should not be confused with the points in (3.47), whose existence was assumed and not proved. Lemma 3.11 is the only place where the hypothesis (3.60) is used in the proof of Theorem 3.10.

Now, as in (3.48), consider the rescaled metrics

g_{i}^{1} = ρ(x_{i}^{1})^{−2} · g_{i}.  (3.64)

Given the hypotheses (3.59) and (3.61), the discussion preceding Proposition 3.6 shows that a subsequence of {g_{i}^{1}} converges on B_{x_{i}^{1}}^{1}(1) to a limit, which is a non super-trivial solution (B^{1}(1), g^{1}, u^{1}, x^{1}) of the static vacuum equations. (With the exception of the non-collapse hypothesis (3.43), this is the only place where the remaining hypotheses, (3.59) and (3.61), are used in the proof of Theorem 3.10). For simplicity, we restrict ourselves in the following to this convergent subsequence. From (3.63), one sees as in Proposition 3.6, c.f.(3.53), that the limit is flat, with u^{1} = const. If necessary, we renormalize {u_{i}} again from now on so that u^{1} ≡ 1 in the limit.

In the following steps, we show that for each i sufficiently large, there is a first k_{o} = k_{o}(i) and points x_{i}^{k_{o}} ∈ U^{k_{o}} such that the sequence of points x_{i}^{k_{o}} are (ρ, c) buffered in the sequence (M, g_{i}), for a specified small c > 0. Here U^{k} and L^{k} are defined as preceding Remark 3.9. This is a quantitative version of the argument in Remark 3.9, and rests basically on the fact that (M, g_{i}, u_{i}) is smooth, for each fixed i.

The construction of such points will take place with a choice for the parameter c for a (ρ, c) buffered sequence, and a choice for a parameter ε measuring the distance of the blow-up metrics to a static vacuum solution. As will be seen in the proof, the choice of c and ε is arbitrary, provided each is sufficiently small, and further ε is sufficiently small, depending on c. While it is possible to make specific numerical choices for c and ε, the work involved is cumbersome and of no specific value, and so we will not do so. The final specification of c and ε will be made in Step IV below, after a preliminary specification in Step II.

From the discussion following (3.64), given an ε > 0, there is an i_{o} = i_{o}(ε) such that for all i ≥ i_{o}, the blow up (B_{x_{i}^{1}}^{1}(1), g_{i}^{1}, x_{i}^{1}) is ε-close to a flat static vacuum limit. By ε-close, we mean that the L^{2} norm of L^{∗}u_{i} on B_{x_{i}^{1}}^{1}(1) is less than ε. Since u_{i} is C^{o} (in fact L^{2,2}) close to 1 at points in B_{x_{i}^{1}}^{1}(1) an arbitrary fixed distance away from the boundary, this implies that the L^{2} norm of the curvature r is less than ε′ = ε′(ε) in balls an arbitrary but fixed distance away from the boundary. In particular, by Theorem 3.4, ρ^{1} is very small, (depending on ε), somewhere near the boundary of B_{x_{i}^{1}}^{1}(1). (Further, by Theorem 3.3, some portion of the level set L^{2} is very close to the boundary of B_{x_{i}}^{1}(1).

Step II. For any given i ≥ i_{o} = i_{o}(ε) fixed, and base points x_{i}^{1} as above, we now inductively construct new base points x_{i}^{2}, x_{i}^{3}, etc, chosen to satisfy quantitative versions of (3.56)-(3.57). For a given choice of c, c_{o} = 10^{−3} and ν_{o} = 10^{−2}, let d = d(c, c_{o}, ν_{o}) be the strong buffer constant determined by c, c_{o} and ν_{o}, as discussed following Lemma 3.8. For the moment, (until Step IV), we require only that c be chosen sufficiently small so that d ≤ \frac{1}{5} and d + 2c ≤ \frac{1}{4}. k^{th}-Level Inductive Hypothesis: For any given i ≥ i_{o}, suppose base points x_{i}^{j} ∈ U^{j}, have been constructed for each j, 1 ≤ j ≤ k, up to a given k ≥ 2. We require that each x_{i}^{j} is not (ρ, c) buffered for j ≤ k − 1, and to be chosen to satisfy the following conditions:

x_{i}^{j} ∈ B_{x_{i}^{j−1}}(ρ(x_{i}^{j−1})),  (3.65)
c · ρ(x_{i}^{j−1}) ≤ dist_{g_{i}}(x_{i}^{j}, ∂B_{x_{i}^{j−1}}(ρ(x_{i}^{j−1}))) ≤ 2c · ρ(x_{i}^{j−1}),  (3.66)

ρ(x_{i}^{j}) ≤ d_{1} · ρ(x_{i}^{j−1}), d_{1} = d + 2c, and  (3.67)
|u_{i}(x) − u_{i}(x_{i}^{j−1})| ≤ \frac{1}{4} · u_{i}(x_{i}^{j−1}), ∀x ∈ B_{x_{i}^{j−1}}((1 − c)ρ(x_{i}^{j−1})).  (3.68)

The conditions (3.65)-(3.66) on the placement of x_{i}^{j}, given x_{i}^{j−1}, can obviously always be satisfied. With regard to the next two conditions, condition (3.68) on the behavior of the potential, and the fact that x_{i}^{j−1} is not (ρ, c) buffered, would imply that x_{i}^{j−1} is not strongly (ρ, d) buffered provided Lemma 3.8 is applicable. If this is the case, it follows that there is a point z_{i}^{j} ∈ B_{x_{i}^{j−1}}(ρ(x_{i}^{j−1})), with ρ(z_{i}^{j}) ≤ d · ρ(x_{i}^{j−1}), and hence by (1.27), there is a point x_{i}^{j} satisfying (3.65)-(3.66) together with (3.67). In other words, (3.67) is a consequence of (3.65), (3.66) and (3.68) to the extent that Lemma 3.8 is applicable. We will show below that indeed it is, even when u_{i}(x_{i}^{j−1}) is very small.

It is important to note that the terminal base point x_{i}^{k} may or may not be (ρ, c) buffered. Further, by the paragraph preceding Step II, there exist base points x_{i}^{2} satisfying (3.65)-(3.68) with k = 2 whenever i ≥ i_{o}, so that one may start the induction process at k = 2.

To normalize the discussion, we scale the metrics g_{i} at each x_{i}^{j} so that ρ_{i}^{j} = 1, i.e. set g_{i}^{j} = ρ(x_{i}^{j})^{−2}·g_{i}, and renormalize u_{i} by setting

u_{i}^{j} = u_{i}/u(x_{i}^{j}).  (3.69)

With this normalization, (3.65)-(3.68) are equivalent to the statements

x_{i}^{j} ∈ B_{x_{i}^{j−1}}^{j−1}(1),  (3.70)
c ≤ dist_{g_{i}^{j−1}}(x_{i}^{j}, ∂B_{x_{i}^{j−1}}^{j−1}(1)) ≤ 2c,  (3.71)
ρ^{j−1}(x_{i}^{j}) ≤ d_{1}, and,  (3.72)
|u_{i}^{j−1}(x) − 1| ≤ \frac{1}{4}, ∀x ∈ B_{x_{i}^{j−1}}^{j−1}(1 − c).  (3.73)

The inductive step is the following: Inductive Step. Suppose x_{i}^{k} is not (ρ, c) buffered. Then there exist base points x_{i}^{k+1} satisfying the conditions (3.65)-(3.68), with k + 1 in place of k.

The next two steps are concerned with the proof of the inductive step. Step III. In order to carry out the inductive step from k to k + 1, we need to analyse the geometry of g_{i}^{k} in the unit ball B_{x_{i}^{k}}^{k}(1). Note that the following Lemma does not assume any buffer condition on x_{i}^{k}, as a sequence in i, for any k fixed.

Lemma 3.12. For any i ≥ i_{o} and some k ≥ 2, suppose base points x_{i}^{k} have been constructed satisfying the k^{th}-level inductive hypothesis. Then the triple (B_{x_{i}^{k}}^{k}(1), g_{i}^{k}, u_{i}^{k}) satisfies the estimates

u_{i}^{k}r_{i} = D^{2}u_{i}^{k} + o(i, k),  Δu_{i}^{k} = o(i, k),  (3.74)

where o(i, k) denotes terms which become arbitrarily small in L^{2}(B_{x_{i}^{k}}^{k}(1)), (in fact in L^{2}(M, g_{i}^{k})), if either i or k is sufficiently large.

Proof. We return to the equation (2.10) for u = u_{i}, i.e.

ur − D^{2}u = −Δu · g + ξ + \frac{s}{3} · g,

applied to g_{i}^{k} in B_{i}^{k}. Divide, i.e. renormalize, this equation by υ_{i,k} ≡ u_{i}(x_{i}^{k}). By (3.66) and (3.68), υ_{i,k} ≥ 2^{−k}. Let u_{i}^{k} = u_{i}/υ_{i,k} as in (3.69), so that u_{i}^{k}(x_{i}^{k}) = 1. This gives

u_{i}^{k}r − D^{2}u_{i}^{k} = −Δu_{i}^{k} · g + υ_{i,k}^{−1} · ξ + υ_{i,k}^{−1} · \frac{s}{3} · g.  (3.75)

Now we claim that the term υ_{i,k}^{−1} · ξ is arbitrarily small in L^{2}, if either i or k is sufficiently large. To see this, recall from Theorem 2.2 that ||ξ||_{L^{2}(M,g_{i})} ≤ C. As noted before in (3.12), in any rescaling g′_{i} = ρ_{i}^{−2} · g_{i}, one has then

||ξ||_{L^{2}(M,g′_{i})} ≤ C · ρ_{i}^{1/2}.  (3.76)

Now by (3.72), in the sequence g_{i}^{j}, for i fixed and any j < k, we have

ρ_{i}^{j} ≤ d_{1} · ρ_{i}^{j−1},

so that

ρ_{i}^{k} ≤ d_{1}^{k} · ρ_{i}^{1}.

It follows that

||υ_{i,k}^{−1}ξ||_{L^{2}(M,g_{i}^{k})} ≤ C · 2^{k} · d_{1}^{k/2}ρ(x_{i}^{1})^{1/2} → 0,

as either i or k → ∞, since d_{1} = d + 2c ≤ 1/4. The same reasoning applies to the last term in (3.75) involving the scalar curvature, since the scalar curvature of (M, g_{i}) is uniformly bounded in L^{∞}. Thus, the last two terms in (3.75) are arbitrarily small, if either i or k is sufficiently large.

Similarly, from Proposition 2.9, the L^{2} norm of Δu is uniformly bounded on (M, g_{i}) and hence again the same reasoning implies that the L^{2} norm of Δu_{i}^{k} goes to 0 uniformly on (M, g_{i}^{k}), as either i → ∞ or k → ∞. Thus the right hand side (3.75) is arbitrarily small whenever either i or k is sufficiently large.

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Lemma 3.12, together with the non-collapse assumption (3.43) implies that for any i ≥ i_{o} = i_{o}(ε) and k ≥ 2, the metrics g_{i}^{k} are ε-close to a static vacuum solution in the sense that the metrics satisfy the static vacuum equations with a (volume normalized) L^{2} error of at most ε. Observe that this ε is the same as the ε chosen initially at the end of Step I. By passing to limits, it follows that in fact g_{i}^{k} on B_{i}^{k} is ε′ = ε‘(ε) close in the weak L^{2,2} topology to an actual static vacuum solution; however, we will not use this fact. By the inductive hypothesis, the predecessors g_{i}^{j}, j ≤ k − 1, are not (ρ, c) buffered at their base points, and so are all δ_{1}-close to a flat metric in B_{i}^{j}(1 − c) in the sense that

(∫_{B_{i}^{j}(1−c)} |r|^{2})^{1/2} ≤ δ_{1};  (3.77)

here δ_{1} = δ_{1}(c, c_{o}) = 2πc · c_{o}. (Again it can be shown that this implies that (B_{i}^{j}(1 − c), g_{i}^{j}) is δ′_{1}-close, δ′_{1} = δ′_{1}(δ_{1}, ν_{o}), in the weak L^{2,2} topology, to a flat metric, but this will not be used). Clearly, δ_{1} may be made arbitrarily small by choosing the initial buffer parameter c sufficiently small. Further, by (3.73), observe that the potential function u_{i}^{j} is \frac{1}{4}-close to a constant function in the C^{0} topology in B_{i}^{j}(1 − c).

Step IV. We are now in position to prove the inductive step. The next result is the main point in this respect. For this, we restrict somewhat further the initial choice of c, and ε. Thus, we require for the remainder of the proof that ε < ε_{o}, where ε_{o} is the constant from Lemma 3.8, and also ε < \frac{1}{10}δ_{1}. In addition, c will be chosen sufficiently small so that d_{1} = d + 2c ≤ \frac{1}{100}, and δ_{2} = 2c_{s} · δ_{1} ≤ \frac{1}{100}, where c_{s} is the Sobolev constant of the embedding L^{2,2} ⊂ C^{0} in ℝ^{3}, (c.f. (3.79) below).

Lemma 3.13. Under the conditions above on c and ε, for any given i ≥ i_{o} = i_{o}(ε), suppose the base point x_{i}^{k} is not (ρ, c) buffered. Assume also that the non-collapse assumption (3.43) holds. Then the potential function u_{i}^{k} is \frac{1}{4}-close to the constant function 1 on B_{x_{i}^{k}}^{k}(1 − c) in the C^{0} topology, i.e. (3.73) holds with k + 1 in place of k.

Proof. The proof is essentially the same as the proof of the similar issue in Remark 3.9, c.f. (3.58). Since x_{i}^{k} is not (ρ, c) buffered and (3.43) holds, the metric g_{i}^{k} is both ε-close to a static vacuum solution, and δ_{1} = δ_{1}(c) close to a flat metric, in the sense defined above, on B^{k}(1 − c) ≡ B_{x_{i}^{k}}^{k}(1 − c).

Let v_{i}^{k} = sup_{B^{k}(1−c)}|u_{i}^{k}|, and recall that u_{i}^{k}(x_{i}^{k}) = 1. It follows from Lemma 3.12 that

(∫_{B^{k}(1−c)} |D^{2}u_{i}^{k}|^{2})^{1/2}≤ (∫_{B^{k}(1−c)}(u_{i}^{k})^{2}|r|^{2})^{1/2}+ε ≤ 2δ_{1} · v_{i}^{k}.  (3.78)

This, together with Sobolev embedding, c.f. [GT, Ch.7], shows that the potential u_{i}^{k} is close in the C^{0} topology to an affine function α = α_{i,k} on B^{k}(1 − c), in that

|u_{i}^{k} − α| ≤ δ_{2}v_{i}^{k}.  (3.79)

The constant δ_{2} is given by δ_{2} = 2c_{s} · δ_{1} ≤ \frac{1}{100}, where c_{s} is the Sobolev embedding constant; c_{s} is a fixed numerical constant in dimension 3, independent of all parameters. This estimate also, of course, applies on domains inside B^{k}(1 − c), with the same α and with v_{i}^{k} replaced by the supremum of |u_{i}^{k}| on the domain. In particular, it follows that sup|u_{i}^{k}| is close to sup|α| on any such domain.

Now to understand more precisely the form of α and the size of v_{i}^{k}, we first consider the validity of (3.79) on the predecessor domain, in the g_{i}^{k} scale. Thus, consider the previous point x_{i}^{k−1}, of distance R_{i}^{k} from x_{i}^{k} in the g_{i}^{k} scale. By (3.71)-(3.72), one computes d_{1}^{−1}(1−2c) ≤ R_{i}^{k} ≤ c^{−1}(1−c), the important point being that R_{i}^{k} is large, since d_{1} is small. Consider also the ball B^{k−1}(R_{i}^{k}) ≡ B_{x_{i}^{k−1}}^{k}(R_{i}^{k}) containing a portion (roughly half) of the ball B^{k}(1 − c). Using the inductive assumption, the L^{2} norm of the curvature of g_{i}^{k} on B^{k−1}(R_{i}^{k}) is much smaller than δ_{1}, since g_{i}^{k} is much larger than g_{i}^{k−1}. This implies that (3.79) holds on B^{k−1}(R_{i}^{k}), i.e.

|u_{i}^{k} − α′| ≤ δ_{2}w_{i}^{k},  (3.80)

where α′ is an affine function on B^{k−1}(R_{i}^{k}) and w_{i}^{k} is the supremum of |u_{i}^{k}| on B^{k−1}(R_{i}^{k}). We may write α′ = a′ + b′t, where t is an affine Euclidean coordinate with t(x_{i}^{k}) = 0.

By the induction hypothesis (3.73), the function

u_{i}^{k} = \frac{u_{i}(x_{i}^{k−1})}{u_{i}(x_{i}^{k})}u_{i}^{k−1}

is 2 · \frac{1}{4} close to the constant function 1 in B^{k−1}(R_{i}^{k}), and hence w_{i}^{k} ≤ \frac{3}{2}. Together with (3.80), this implies that α′ = a′ + b′t satisfies |b′| ≤ (R_{i}^{k})^{−1} < 2d_{1}. Thus the oscillation of α′ is small. Further |1 − α′| ≤ 2δ_{2}.

It follows that on the ’half-ball’ D_{i}^{k} ≡ B^{k}(1 − c) ∩ B^{k−1}(R_{i}^{k}), the function u_{i}^{k} satisfies

|u_{i}^{k} − 1| ≤ 2(δ_{2} + d_{1}).  (3.81)

Now return to the estimate (3.79). It follows from (3.81) that sup_{D^{k}}|u_{i}^{k}| is close to 1. Hence, by (3.79) and the discussion in the paragraph following it, one may calculate that v_{i}^{k} = sup_{B^{k}(1−c)}|u_{i}^{k}| ≤ 5. It then follows from the estimates (3.78)-(3.79) applied to the union B^{k}(1 − c) ∪ B^{k−1}(R_{i}^{k}) that α and α′ are close. Working out concretely the degree of closeness then gives the estimate

|u_{i}^{k} − 1| ≤ 10(δ_{2} + d_{1}),  (3.82)

on B^{k}(1 − c). The result then follows from the specifications of δ_{2} and d_{1} above.

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We are now in position to argue the inductive step.

Lemma 3.14. Suppose x_{i}^{k} satisfies the k^{th}-level inductive hypothesis (3.65)-(3.68). If the base point x_{i}^{k} is not (ρ, c) buffered, then there exist points x_{i}^{k+1} satisfying the (k + 1)^{st}-level inductive hypothesis, i.e. (3.65)-(3.68) with k + 1 in place of k everywhere.

Proof. Given the previous work, this is now essentially obvious. There are clearly (many) base points x_{i}^{k+1} satisfying (3.65)-(3.66) with k + 1 in place of k. Since x_{i}^{k} is not (ρ, c) buffered, by Lemma 3.13 u_{i}^{k} is \frac{1}{4}-close to the constant function 1 in B_{x_{i}^{k}}^{k}(1 − c), so that (3.68), (or (3.73)) holds, with k + 1 in place of k. By Lemma 3.12, or more precisely its contrapositive, x_{i}^{k} is not strongly (ρ, d) buffered, and so there exist x_{i}^{k+1} such that (3.67) also holds, with k + 1 in place of k.

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Step V. Now, since (M, g_{i}) is a smooth compact Riemannian manifold, ρ(M, g_{i}) cannot be arbitrarily small for any given i ≥ i_{o} i.e. there exists ρ_{o} = ρ_{o}(i) such that ρ(M, g_{i}) ≥ ρ_{o} > 0. Starting with the initial sequence x_{i}^{1} from Lemma 3.11, and any i ≥ i_{o}, the inductive process above constructs points x_{i}^{k}, k ≥ 2, satisfying ρ(x_{i}^{k}) ≤ d_{1}ρ(x_{i}^{k−1}), and continues indefinitely as long as x_{i}^{k} is not (ρ, c) buffered. It follows that there is a first level k_{o} = k_{o}(i) such that y_{i} ≡ x_{i}^{k_{o}} is (ρ, c) buffered. Observe that the discussion in Remark 3.9 proves that necessarily k_{o}(i) → ∞ as i → ∞. The sequence (B_{i}^{k_{o}}(1), g_{i}^{k_{o}}, y_{i}, u_{i}^{k_{o}}) satisfies the equation (3.75), with the right side going to 0 uniformly in L^{2}(M, g_{i}^{k_{o}}) as i → ∞. Thus, (c.f. the discussion following Definition 3.7), a subsequence converges to a non-flat limit solution (D, g′) to the static vacuum equations, with (possibly renormalized) potential function ū = lim u_{i}^{k_{o}}. (Here we are using again the non-collapse assumption (3.43)). The limit domain D may be chosen for instance to be an open domain in the limit B_{y′}(1) of B_{i}^{k_{o}}(1) which has smooth and compact closure in B_{y′}(1). This completes the proof of Theorem 3.10.

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We conclude this subsection with a number of remarks.

Remark 3.15. (i). At this point, if not well before, one may wonder why not just blow-up the sequence {g_{i}} at base points {q_{i}} realizing the minimum ρ(M) of ρ(x), x ∈ {(M, g_{i})}, in which case {q_{i}} is strongly (ρ, 1) buffered. In this case, a limit of the metrics

g′_{i} = ρ(M)^{−2} · g_{i}

will automatically be complete and at least L^{2,2} smooth.

The main problem is that, from the work above, there is every indication that this minimum occurs at, or arbitrarily near the 0-levels of u. The z-splitting of {g′_{i}} has the form

u_{i}z′_{i} = (D^{2})′u_{i} − Δ′_{i}u_{i} + ξ′_{i} − \frac{s′_{i}}{3} · g′_{i}.

Although, as before, the last three terms on the right go to 0, it may well happen that u_{i} converges to 0 uniformly when blowing up at {q_{i}}, so that in the limit this equation gives just 0 = 0, i.e. a supertrivial solution, and thus no information whatsoever regarding the limit metric. Examples exhibiting exactly this behavior will be discussed in §6. Further, without the descent construction as in the proof of Theorem 3.10, there may be no means to renormalize u to obtain a non-trivial solution.

(ii). It is clear that all the preceding arguments are unchanged if u is replaced by −u, so that the construction of Theorem 3.10 may be carried out either “down” the u-levels from a local maximum, as in (3.63), or “up” the u-levels from a local minimum, provided such a local minimum is bounded away from zero.

(iii). We point out that it is not possible in the construction above to descend down to other levels besides the 0-level, e.g. trying to descend only to the level u = \frac{1}{2} by renormalizing the differences \frac{1}{2} + 2^{−k−1} and \frac{1}{2} + 2^{−k}. This is due essentially to the linear, and not affine, nature of the equation (3.71) in u.

(iv). The construction in Theorem 3.10 has been referred to as a ‘descent’ construction down the levels of u = u_{i}, since this is what occurs in all known examples, (as in Example 3.9), and since one certainly descends to u-levels less than max u_{i}/T_{i} = 1. However, the construction itself does not guarantee such a descent, i.e. that

u_{i}(x_{i}^{j}) < u_{i}(x_{i}^{j−1}) for j ≤ k_{o}.  (3.83)

Only the control (3.68) or (3.73) is obtained on the behavior of u.

On the other hand, although it has not been used in the proof, it is worth noting that at any stage in the inductive construction, the level sets L^{s} of u_{i}^{j}, j < k_{o}, for any given s with 0 < s < 10^{−1}, are somewhere δ = δ(d) close to ∂B_{x_{i}^{j}}^{j}(1) in the g_{i}^{j} metric; here δ may made arbitrarily small by choosing

d = d(c) sufficiently small. This follows from the fact that x_{i}^{j} is not (ρ, c) buffered, and hence not strongly (ρ, d) buffered, by Lemma 3.8.

Using this remark, it is possible to define variants of the inductive construction in Theorem 3.10 for which (3.83) does hold, and for which the same conclusions are valid. Such variations will not be used here however.

Remark 3.16. (i). The construction in Theorem 3.10 is very local. Thus, given any initial sequence {x_{i}^{1}} satisfying the conclusions of Lemma 3.11, there is a buffered sequence {y_{i}} satisfying the conclusions of Theorem 3.10 very close to x_{i}^{1}; in fact dist_{g_{i}}(x_{i}^{1}, y_{i}) ≤ 2ρ(x_{i}^{1}).

However, there may be many possible (essentially) distinct choices of the initial sequence {x_{i}^{1}} in (M, g_{i}) satisfying (3.60) and (3.63). Note also that the choice of y_{i} is by no means unique, since the choice of the intermediate points x_{i}^{j} may allow considerable freedom.

(ii). The limit solution (D, g′) in Theorem 3.10 is only locally defined, and further may live in the region of M where u is approximately 0, since the potential function ū is obtained by renormalizing, possibly infinitely many times, the original sequence {u_{i}}. The geometry of the limit solutions (D, g′) will be studied in more detail in §5.

Remark 3.17. The proof of Theorem 3.10 requires Lemma 3.11 to obtain the initial sequence {x_{i}^{1}} whose blow-up limit is a flat static vacuum solution, with limit potential function u^{1} = const. > 0. On the other hand, if one is given some initial sequence {p_{i}^{1}} satisfying this latter property, (not necessarily satisfying the local maximum condition (3.63)), then the construction in Theorem 3.10 may be applied starting at {p_{i}^{1}}, without any further changes, to reach the same conclusions.

There are instances where the hypothesis (3.60) in Theorem 3.10 can be deduced or replaced by suitable assumptions on the local behavior of the functions u_{i}. We give two examples of this below.

Proposition 3.18. Let (M, g_{i}) be a sequence of unit volume Yamabe metrics satisfying the noncollapse assumption (3.43). Suppose there exists a δ > 0, a sequence r_{i} → 0, and a sequence of base points {x_{i}} ∈ (M, g_{i}), such that on B_{i} = B_{x_{i}}(r_{i}),

osc_{B_{i}} (u_{i}/T_{i}) ≥ δ.  (3.84)

Then (3.60) holds, with δ in place of δ_{o}.

Proof. It suffices to show that ρ(x_{i}) → 0 as i → ∞. If there exists ρ_{o} > 0 such that ρ(x_{i}) ≥ ρ_{o}, (in a subsequence), then L^{2} estimates on the trace equation (2.13), c.f. [GT, Thm.8.8], imply that {u_{i}/T_{i}} is uniformly bounded in L^{2,2}(B_{x_{i}}(ρ_{o}/2)). Sobolev embedding then implies that u_{i}/T_{i} is uniformly bounded in C^{1/2}, so that in particular the oscillation of u_{i}/T_{i} on B_{x_{i}}(r) is bounded by r^{1/2}. This contradicts (3.84).

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Somewhat more generally, the hypotheses (3.60) and (3.61) in Theorem 3.10 can be replaced by the following local condition on the behavior of u = u_{i} near its 0-level set. Let t(x) = dist_{g_{i}}(x, L^{o}). If u_{i} > 0 everywhere, L^{o} must be replaced by L^{ε_{i}}, for a suitable sequence ε_{i} → 0.

Proposition 3.19. Let (M, g_{i}) be a sequence of unit volume Yamabe metrics satisfying (3.59) and the non-collapse assumption (3.43). Suppose there exists a sequence of base points {x_{i}} ∈ (M, g_{i}), with t(x_{i}) → 0 as i → ∞, such that

u_{i}(x_{i}) ≫ t^{1/2} as i → ∞,  (3.85)

and

t|∇logu_{i}|(q_{i}) < \frac{1}{2},  (3.86)

for all q_{i} ∈ B_{x_{i}}(K_{i}t(x_{i})) satisfying \frac{1}{2}t(x_{i}) ≤ t(q_{i}) ≤ K_{i}t(x_{i}); here K_{i} is any given sequence with K_{i} → ∞ as i → ∞ and it is assumed that t achieves the value K_{i}/2 in B_{x_{i}}(K_{i}t(x_{i})). Then the conclusions of Theorem 3.10 hold for some (ρ, c) buffered sequence y_{i} with dist_{g_{i}}(x_{i}, y_{i}) → 0 as i → ∞.

Proof. First, note that as in Proposition 3.18, (3.85) forces ρ(x_{i}) → 0, since if ρ(x_{i}) were bounded away from 0, L^{2} elliptic estimates applied to the trace equation (2.13) as above imply that u_{i} is bounded in L^{2,2} ⊂ C^{1/2}, which violates (3.85).

Consider the behavior of the pointed sequence (M, g′_{i}, x_{i}, ū_{i}), g′_{i} = t(x_{i})^{−2} ·g_{i}. We claim that Lemma 3.12 holds on B′_{x_{i}}(1). Thus, for ū_{i} = u_{i}/u(x_{i}), (compare with (3.75)), the renormalized equation for u takes the form

ū_{i}r − D^{2}ū_{i} = −Δū_{i} · g + u(x_{i})^{−1} · ξ + u(x_{i})^{−1} · \frac{s}{3} · g.  (3.87)

As in (3.76), in this scale, we have

u(x_{i})^{−1}||ξ||_{L^{2}} ≤ C · u(x_{i})^{−1}t(x_{i})^{1/2} → 0, as i → ∞,

where the last estimate follows from (3.85). The same reasoning as in Lemma 3.12 shows that the full right side of (3.87) also goes to 0, as i → ∞, which proves the claim.

Next we claim that there exists a constant ρ_{o} > 0 such that

ρ(x_{i}) ≥ ρ_{o} · t(x_{i}).  (3.88)

For in the scale g′_{i} where t′(x_{i}) = 1, note that ū_{i}(x_{i}) = 1, and by (3.86), ū_{i} is bounded away from 0 in the region where \frac{1}{2} ≤ t′ ≤ 2; of course here t′(p) = dist_{g′_{i}}(p, L^{o}). Hence, (3.88) follows from Theorem 3.3 applied to the triple (M, g′_{i}, ū_{i}). Since ρ′(x_{i}) is bounded away from 0, blow-up limits based at x_{i} exist. Further, the preceding discussion shows that all blow-up limits based at x_{i} converge to a static vacuum solution with potential function ū.

Now if a blow-up limit (M, g′_{i}, x_{i}) happens to be non-flat, we are done (the assertion is proved in this case). Suppose instead that the blow-up limit is flat. Then the limit function ū is either a non-constant affine function or ū = const. = 1. Hence it suffices to prove that the latter case holds, since then all the arguments following Lemma 3.11 can be carried over without change, c.f. Remark 3.17.

Suppose the limit ū is a non-constant affine function. The product (3.86) is scale-invariant, and hence on the limit, one obtains

t′|∇logū|(q) ≤ \frac{1}{2},

for all q in B′_{x_{i}}(K_{i}) satisfying t′(q) ≥ \frac{1}{2}. The assumption on t and K_{i} implies that on the limit, t′ is an unbounded function. However, it is obvious that no non-constant affine function satisfies this bound.

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To conclude this subsection, it is clear that Theorem 3.10 proves Theorem A(II) in case σ(M) < 0, or more precisely in case the scalar curvature of {g_{i}} satisfies (3.59).

In the following two subsections, we complete the proof of Theorem A(II) and Theorem 3.10 in the cases σ(M) = 0 and σ(M) > 0. As will be seen, this requires some, but as it turns out, no truly essential changes.

§3.5. Suppose

σ(M) = 0,

or more precisely suppose {g_{i}} is a sequence of unit volume Yamabe metrics on M satisfying

s_{g_{i}} → 0.  (3.89)

As before, we require that the assumptions (3.42) and (3.43) hold uniformly on {g_{i}}.

Of course, if λ does not converge to 0, so that (3.45) holds, then all of the results of §3.4 still hold, with identical proofs, at least when s_{g_{i}} ≤ 0. However, as discussed in Remark 2.12, one often expects that λ → 0 when (3.89) holds. In fact, if σ(M) > 0 and s_{g_{i}} → 0, then as discussed following Proposition 2.9, there are many unit volume Yamabe metrics g_{i} which do not degenerate at all, and for which f → −1, u → 0, and ξ → 0 smoothly.

Thus, suppose that λ = λ_{i} → 0 on (M, g_{i}). Assuming {g_{i}} degenerates in the sense of (0.8), since u may go to 0 in L^{2}, one might not expect to produce any non-trivial blow-up solutions of the static vacuum equations by descent down the u-levels. However, when λ → 0, we also have δ = − ∫ k → 1, by (2.17) and (2.28). In particular, sup(−k) is uniformly bounded away from 0 on {g_{i}}, compare with (3.61).

Given these preliminaries, we prove Theorem 3.10 with the assumption (3.89) replacing (3.59).

Proof of Theorem 3.10: σ(M) = 0.

We assume that {g_{i}} is a degenerating sequence of unit volume Yamabe metrics on U_{δ_{o}} ⊂ M , satisfying (3.89), together with the L^{2} bound on z^{T} (3.42), (or (0.19)), and the non-collapse assumption (3.43). For simplicity, assume first that

s_{g_{i}} ≤ 0.  (3.90)

The splitting equation (2.25) for z^{T} may be written in the form

kr = D^{2}k − Δk · g − z^{T} + ξ.  (3.91)

By (3.42), (and also (2.11)), the term z^{T} −ξ is uniformly bounded in L^{2}, as is the term Δk, (by taking the trace of (2.25)).

This equation, and its associated trace equation, now have exactly the same form and bounds as the u-equation (2.10), with −k in place of u and z^{T} − ξ in place of ξ + \frac{s}{3}g. As noted above, the mean value of −k, namely δ, converges to 1 as i → ∞.

Exactly the same analysis as in §3.3 and §3.4 may now be carried out on the function −k, or more precisely −k/ sup(−k) in place of u/ sup u. Note that from (2.28),

\frac{−k}{sup(−k)} = \frac{u}{sup u},  (3.92)

so that the main hypothesis (3.60) holds with −k in place of u. Similarly, analogous to (3.61), sup(−k) is uniformly bounded away from 0, since its mean value converges to 1. Thus, one produces a (ρ, c) buffered sequence with associated limit giving a non-flat static vacuum solution, with limit potential function k̄ coming from the geometry of {k_{i}}. In fact, by (3.92), the potential function can also be obtained from renormalizations of the {u_{i}}.

The proof in case

s_{g_{i}} ≥ 0, and s_{g_{i}} → 0,  (3.93)

is exactly the same as in the case (3.90) with one exception. Namely, if s_{g_{i}} > 0, it is possible that Ker L^{∗} ≠ 0 on (M, g_{i}) so that the functions u or k may not be uniquely defined. In this situation, one just takes any choice for −k in {−k + Ker L^{∗}}. Since the choice is always renormalized by its supremum, this has no effect on any of the arguments. This situation will be discussed in more detail in §3.6, where it occurs more naturally.

This completes the proof of Theorem 3.10 and Theorem A in case σ(M) = 0, i.e. (3.89) holds. ■

§3.6. Finally, we turn to the case where

σ(M) > 0,

or more generally where {g_{i}} is a sequence of unit volume Yamabe metrics with

s_{g_{i}} ≥ s_{o} > 0.  (3.94)

As noted in §2, such metrics may satisfy

Ker L^{∗} ≠ 0.  (3.95)

For instance, on the standard sphere S^{3}, the 1^{st} eigenfunctions of the Laplacian form exactly Ker L^{∗}, see also §6.4. Note that the condition (3.95) implies, by taking the trace, that

−s/2 ∈ Spec Δ.  (3.96)

The condition (3.95) is equivalent to the statement that the linearization L of s at a Yamabe metric g is not surjective onto C^{∞}(M, ℝ), and has been studied by several authors, c.f. [Bo], [Kb2], [La]. In particular, the last two authors show that there are many (even conformally flat) Yamabe metrics satisfying (3.95) which are not Einstein.

Nevertheless, even for metrics satisfying (3.95), all the splittings (2.6), (2.10) (2.21) and (2.25) are valid and defined as before. The associated functions f, u and k however are obviously not uniquely defined; they are unique only modulo Ker L^{∗}. Related to this, as noted following Proposition 2.9, there is no apriori bound on the T^{2,2} norm or even the L^{2} norm of f.

On the other hand, if g is a Yamabe metric with s_{g} > 0, then by the definition of g - as realizing inf S in the conformal class [g] - one has

s_{g} · (∫ ψ^{6}dV_{g})^{1/3} ≤ ∫ (8|dψ|^{2} + sψ^{2}) dV_{g},  (3.97)

for any positive smooth function ψ on M, cf. [Bes,4.28], [LP]. It follows that (3.97) holds for all smooth functions ψ on M. Thus on (M, g) one has a Sobolev inequality, uniform on any sequence of Yamabe metrics for which s_{g} > 0 is uniformly bounded away from 0.

This has two immediate consequences. First, we have:

Lemma 3.20. Let (M, g_{i}) be a sequence of unit volume Yamabe metrics on M satisfying (3.94). Then there is a constant ν_{o} = ν_{o}(s_{o}) such that the volume radius satisfies

ν_{i}(x) ≥ ν_{o},  (3.98)

for all x ∈ (M, g_{i}). Thus, the sequence {g_{i}} cannot collapse anywhere, c.f. §1.4.

Proof. It is a standard fact that the Sobolev inequality (3.97) gives rise to a uniform lower bound on the volumes of geodesic balls in comparison to the volumes of Euclidean balls, i.e. to the ratio vol(B_{x}(r))/r^{3}, provided vol B_{x}(r) < \frac{1}{2} vol(M, g_{i}); we refer to [Ak] for example for a proof. By definition of the volume radius, this gives (3.98). Similarly, it follows from the definition of collapse in §1.4 that (3.98) prevents collapse of the sequence (M, g_{i}) at any sequence of base points.

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Hence, in the case of uniformly positive scalar curvature, condition (i) of Theorem A is automatically satisfied.

Second, from the trace equation (2.12),

2Δf + sf = tr ξ,  (3.99)

and the fact that the right side is uniformly bounded in L^{2}, see Theorem 2.2, it follows from standard elliptic estimates using only the Sobolev inequality (3.97), c.f. [GT, Thm.8.15], that

sup |f| ≤ c(s_{o}) {(∫ f^{2} dV_{g})^{1/2}) + 1} .  (3.100)

Of course, the T^{2,2} norm of f is also bounded by the L^{2} norm of f; see the proof of Proposition 2.9.

We now complete the proof of Theorem 3.10, with (3.94) in place of (3.59).

Proof of Theorem 3.10: σ(M) > 0.

Assume that {g_{i}} is a degenerating sequence of unit volume Yamabe metrics on U_{δ_{o}} ⊂ M, satisfying (3.94), together with the L^{2} bound on z^{T} (3.42), (or (0.19)). By Lemma 3.20, no non-collapse assumption is needed.

As noted in Remark 2.12, the bound (3.94) implies that λ = λ_{i} is bounded away from 0, and hence T_{i} = sup u_{i} ≥ λ_{o}, for some λ_{o} > 0; thus one has the bound (3.61) in this context.

If Ker L^{∗} = 0 for all g_{i}, then all of the previous arguments in §3, in particular the proof of Theorem 3.10, hold without any changes.

Suppose instead that Ker L^{∗} ≠ 0 on some subsequence of {g_{i}}. In this case, simply make some choice for the function u = u_{i} mod Ker L^{∗}. Note that λ = ∫ u is independent of any choice. As indicated above in §3.5, recall that throughout the previous work in §3, the potential u is initially renormalized by its maximum, i.e. only the behavior of u/sup u is considered. By (3.100) this renormalization is equivalent to the renormalization of u by its L^{2} norm. Of course if ||u_{i}||_{L^{2}} → ∞, then the initial structural equation (2.10) becomes

L^{∗} (\frac{u_{i}}{||u_{i}||_{L^{2}}} ) → 0, in L^{2}(M, g_{i}), as i → ∞,

which is even stronger than the initial L^{2} bound on L^{∗}(u), compare with the renormalization in the proof of Proposition 3.1. The proofs of all the previous results in §3 follow exactly as before, without any changes.

This completes the proof of Theorem A and Theorem 3.10 in all cases.

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Remark 3.21. We point out that the assumption that the metrics {g_{i}} in Theorem A or Theorem 3.10 are defined on a fixed 3-manifold M has not been used. In fact, these and all previous results hold under the same assumptions on arbitrary sequences of closed oriented Riemannian 3-manifolds (M_{i}, g_{i}).

4. Remarks on the Hypotheses.

In this section, we make some further comments on the hypotheses of Theorem A. First, we consider the non-collapse hypothesis (i) in Theorem A, i.e. (0.18) or more generally (3.43), and afterwards the degeneration hypothesis (iii) in Theorem A, i.e. (0.20). The hypothesis (ii), i.e. the L^{2} bound on z^{T} is already discussed in Theorem 2.10-Remark 2.12.

§4.1. By Lemma 3.20, the non-collapse assumption (3.43) is only needed in the case σ(M) ≤ 0. From some perspectives however, this assumption is perhaps the most crucial in Theorem 3.10 or Theorem A.

The basic difficulty in handling the case of collapse as opposed to convergence is already seen in the discussion in §3.2, for instance in the proof of Proposition 3.1. Thus, consider a blow-up sequence g′_{i} = ρ(x_{i})^{−2} · g_{i}, with ρ(x_{i}) → 0. If this sequence collapses, that is

ν(x) << ρ(x),  (4.1)

then it can be proved that the collapse is along a sequence of injective F-structures on B′_{x_{i}}(1 − ε_{i}), for some sequence ε_{i} → 0 as i → ∞. We may then unwrap the collapse, (i.e. resolve the degeneration), by considering the universal covers B̃′_{x_{i}}(1 − ε_{i}). This sequence does not collapse anywhere. Thus, a subsequence of (B̃′_{x_{i}}(1 − ε_{i}), g_{i}, x_{i}) converges to a limit L^{2,2} Riemannian manifold (B̃′_{x}(1), g′, x); see [An2, §3] for further details here.

The L^{2} curvature radius is essentially invariant under coverings. However, in the process of passing to the universal cover, we lose the property that

∫ |ξ′_{i}|^{2}dV′_{i} = ρ(x_{i}) · ∫ |ξ_{i}|^{2}dV_{i} → 0,  (4.2)

see (3.12). In passing to the universal cover, the apriori L^{2} bound (2.11) on ξ is lost. The property (4.2) was crucial in the case of convergence. The fact that ξ′ → 0 in L^{2} as in Proposition 3.1 implies that the limit is a solution of the static vacuum Einstein equations. Without this property, the limit does not satisfy any particular equation.

Recall that, under the assumption (3.43),

ν_{i} ≥ ν_{o} · ρ_{i},  (4.3)

the previous results of §3 hold in U_{δ_{o}} for an arbitrary sequence of Yamabe metrics {g_{i}} with a bound on the L^{2} norm of z^{T} . Even though the sequence {g_{i}} satisfies no particular equation (PDE), (besides being a Yamabe metric), the blow-up limits modeling degenerations of {g_{i}} do satisfy a strong PDE, namely the static vacuum equations. It seems quite unreasonable to believe this without some assumption like (4.3). Note that the integral bound ∫ |ξ|^{2} ≤ \frac{s^{2}}{3} from (2.11), which gives rise to (4.2) in blow-ups, becomes less and less meaningful in regions where (M, g_{i}) is more and more collapsed, since the bound may then come primarily from the volume collapse and not reflect any particular behavior of ξ itself.

This difficulty can be overcome if one had apriori L^{∞} control on ξ in place of an L^{2} bound, (since such bounds are invariant under coverings), or if the sequence of Yamabe metrics satisfies some other (stronger) P.D.E. This latter will be the path taken in [AnII].

§4.2. Next we make some remarks on the hypothesis (iii) of Theorem A. Suppose {g_{i}} is a degenerating sequence of unit volume Yamabe metrics on M, so that ||z_{g_{i}}||_{L^{2}(M)} → ∞. It is natural to ask if then (iii) is satisfied. In general, that is for arbitrary sequences, the answer is no, see §6 for further discussion.

Suppose for instance {g_{i}} is a unit volume maximizing sequence for the functional v^{2/3} · s, so that

s_{g_{i}} → σ(M).  (4.4)

We will see in §7, c.f. Theorem 7.2 and Lemma 7.4, that any such sequence can be perturbed slightly if necessary, for instance in the T^{2,2} topology, so that the sequence is still maximizing, and so that the gradient −z^{T} = ∇(v^{2/3} · s|_{C}) goes to 0 in the natural dual topology, i.e. at g_{i},

||z^{T} ||_{T^{−2,2}(T C)} = sup {|∫_{M}⟨z^{T} , α⟩dV | : α ∈ T C and ||α||_{T^{2,2}} ≤ 1} → 0,  (4.5)

as i → ∞. In other words, the sequence {g_{i}} is Palais-Smale for the functional S|_{C} w.r.t. the T^{2,2} metric on 𝕄. In particular, there are many such Palais-Smale sequences.

Consider now sequences {g_{i}} ∈ C_{1} which satisfy the stronger condition that

||z^{T} ||_{T^{−2,2}(T 𝕄)} → 0;  (4.6)

in other words, the condition (4.5), but where α is no longer constrained to lie in T C. Such sequences will be called strongly Palais-Smale. It follows easily from the definition that then also

tr z^{T} → 0 in T^{−2,2}.  (4.7)

Under these circumstances, one has the following result.

Proposition 4.1. Let {g_{i}} be a sequence of unit volume Yamabe metrics satisfying the estimate (4.7), and suppose σ(M) < 0, or just s_{g_{i}} ≤ s_{o} < 0. Then for the associated functions u = u_{i}, one has

||\frac{u−λ}{λ} ||_{L^{2}} → 0, as i → ∞,  (4.8)

where the L^{2} norm is w.r.t. the metric g = g_{i}.

Proof. Consider the trace equation (2.13) for u, that is, using (2.33),

2Δ (u − λ) + s (u − λ) = λ tr z^{T} .  (4.9)

Set v = \frac{u−λ}{λ} and let ϕ = ϕ_{i} be the solution on (M, g) = (M, g_{i}) to the equation

Δϕ − ϕ = \frac{v}{||v||_{L^{2}}} .  (4.10)

Since 1 is not in the spectrum of Δ, (Δ is a non-positive operator), this equation has a unique solution. To estimate the T^{2,2} norm of ϕ, pair equation (4.10) with ϕ and integrate by parts to obtain

∫ |dϕ|^{2} + ∫ ϕ^{2} = − ∫ ϕ\frac{v}{||v||_{L^{2}}} ≤ (∫ ϕ^{2})^{1/2} .  (4.11)

Further, squaring both sides of (4.10) gives

∫ (Δϕ)^{2} + 2|dϕ|^{2} + ϕ^{2} = 1.  (4.12)

Thus, the T^{2,2} norm of ϕ is bounded by 1.

It follows that ϕ is an admissible test function in (4.7), so that from (4.9), one obtains

∫ 2ϕΔv + ϕsv → 0.

Integrating by parts gives

∫ 2v [\frac{v}{||v||_{L^{2}}} + ϕ] + ϕsv → 0,

or,

2||v||_{L^{2}} + ∫ (2 + s)ϕv → 0.  (4.13)

But from (4.10)

||v||_{L^{2}} (∫ |dϕ|^{2} + ∫ ϕ^{2}) = − ∫ ϕv,

so that,

2||v||_{L^{2}} − (2 + s)||v||_{L^{2}}||ϕ||_{L^{1,2}} → 0.

This implies

||v||_{L^{2}}(2 − (2 + s)||ϕ||_{L^{1,2}}) → 0.  (4.14)

Thus either ||v||_{L^{2}} → 0, as required, or

||ϕ||_{L^{1,2}} → \frac{2}{2 + s}.  (4.15)

This of course implies s > −2, and since σ(M) < 0, \frac{2}{2+s} > \frac{2}{2+σ(M)} > 1, (for all i), which with (4.15) contradicts (4.11). Hence the result follows.

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Note that this argument makes strong use of the T^{2,2} norm; it is not (likely to be) valid w.r.t. the L^{2,2} norm. It is quite easy to see that there are counterexamples to Proposition 4.1 when σ(M) > 0, c.f. §6.4. If σ(M) = 0, or more precisely {g_{i}} is a sequence of Yamabe metrics with s_{g_{i}} → 0, then this result is borderline. Namely, the same argument as above gives either the conclusion (4.8), or (from (4.15), ||dϕ||_{L^{2}} → 0 and ||ϕ||_{L^{2}} → 1, with ∫ ϕ = 0. The latter case is of course impossible if λ_{1}, the lowest non-zero eigenvalue of Δ, is bounded away from 0, but without some assumption of this kind, it is not clear if (4.8) follows from (4.7).

Now in case (4.8) holds, it is easy to see that condition (iii) of Theorem A follows from the noncollapse assumption (0.18) or (3.43) and the general degeneration assumption (0.8), (in place of (0.20)). In fact, (compare with Lemma 3.11 and (3.47)), we have

Proposition 4.2. Let (M, g_{i}) be a sequence of unit volume Yamabe metrics satisfying the non-collapse assumption (3.43) together with (4.8).

If {g_{i}} degenerates on M, i.e. if there are points p_{i} ∈ (M, g_{i}) such that ρ(p_{i}) → 0, then there are points x_{i} ∈ (M, g_{i}) such that

|u_{i}(x_{i})/T_{i} − 1| → 0, as i → ∞,  (4.16)

and

ρ(x_{i}) → 0, as i → ∞,  (4.17)

where T_{i} = sup u_{i}.

Proof. Note that given (4.8), we do not assume σ(M) < 0. Note also that u/T = (\frac{u}{λ})/(\frac{T}{λ}).

First, suppose T/λ = T_{i}/λ_{i} → ∞. We claim that any sequence {x_{i}} satisfying (4.16) must then satisfy (4.17). For if not, then for some ρ_{o} > 0 and for some sequence x_{i} as above,

ρ(x_{i}) ≥ ρ_{o}.  (4.18)

From (4.18) and the non-collapse assumption, it follows that B = B_{x_{i}}(ρ_{o}) has L^{2,2} bounded geometry. Thus, by Theorem 2.10, z^{T} is uniformly bounded in L^{2}. Consider the trace equation (4.9),

2Δ(\frac{u − λ}{λ}) + s(\frac{u − λ}{λ}) = tr z^{T} .  (4.19)

The right side of (4.19) is uniformly bounded in L^{2}, so that L^{2} elliptic estimates [GT, Thm.8.8] imply a bound

||v||_{L^{2,2}(D)} ≤ C[||v||_{L^{2}(B)} + 1],  (4.20)

for D ⊂⊂ B and v = \frac{u−λ}{λ}. The right side of (4.20) is bounded by (4.8) so that Sobolev embedding on B implies that sup u/λ = T/λ is bounded. Thus, under the assumptions above, (4.18) cannot hold.

The same argument as above holds if only lim sup T_{i}/λ_{i} > 1. Namely, this assumption together with (4.20) and Sobolev embedding then implies that, in some subsequence, u/λ = u_{i}/λ_{i} ≥ 1 + ε, for some ε > 0, on a ball D ⊂ B whose volume is uniformly bounded below, if (4.18) held. This is impossible since u_{i}/λ_{i} → 1 in L^{2} by (4.8). Thus, in the following, we may assume

lim T_{i}/λ_{i} = 1.  (4.21)

For a given t < 1, let U_{i} = U_{i}(t) = {x ∈ (M, g_{i}) : u_{i}(x)/λ_{i} ≥ t}. Suppose that there were ρ_{o} > 0 such that

ρ(x) > ρ_{o}, ∀x ∈ U_{i}.  (4.22)

Since u_{i}/λ_{i} → 1 in L^{2}, it follows that vol(M \ U_{i}) → 0. Then, arguing as above on (4.19), one also has u_{i}/λ_{i} → 1 pointwise on compact subsets of each B = B_{x_{i}}(ρ_{o}), x_{i} ∈ U_{i}. Observe that these balls must cover M. For if not, then there exist balls B_{q_{i}}(ρ_{o}) disjoint from U_{i}. On the other hand, by the noncollapse assumption (3.43), such balls have a definite volume, and hence B_{q_{i}}(ρ_{o}) must have intersected U_{i}. It follows (from (1.27)), that

ρ(x) ≥ ρ_{o}/2, ∀x ∈ M,

which contradicts the assumption ρ(p_{i}) → 0, for some p_{i} ∈ (M, g_{i}). This means that (4.22) does not hold, so that for all t < 1, there exist x_{i}(t) ∈ U_{i}(t) such that ρ(x_{i}(t)) → 0 as i → ∞. Since t is arbitrary, one may choose a sequence t_{i} → 1 suitably slowly as i → ∞ to give (4.16) and (4.17).

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We note that Proposition 4.2 does not require λ to be bounded away from 0, or any bound on the L^{2} norm of z^{T} .

Also, note that if T_{i}/λ_{i} → ∞, then the proof above does not require the hypothesis (4.8), at least when σ(M) < 0 Hence, in this case, the main hypothesis (3.60) of Theorem 3.10 is automatically satisfied if {g_{i}} degenerates somewhere on M.

Although the estimate (4.8) follows from (4.7) when σ(M) < 0, it is not clear how to construct, on general manifolds M, sequences of Yamabe metrics satisfying (4.6) or (4.7), even though the Palais-Smale condition (4.6) is easily realized. In §6.5, we construct examples of sequences satisfying (4.6); in fact these sequences satisfy much stronger conditions, c.f. also the discussion in §7.

Remark 4.3. We will not focus here on any applications of Theorem 3.10, but mention one kind of potential application. Suppose {g_{i}} is a sequence of unit volume Yamabe metrics, for instance a maximizing sequence, satisfying the conditions of Theorem A(II). Suppose further that it can be proved on other (geometric or topological) grounds that there are no non-trivial static vacuum solutions arising from blow-ups g′_{i} = ρ(x_{i})^{−2} · g_{i}. (As an example of this, we mention that it can be shown that there are no blow-up limits (N, g_{o}) which are non-trivial static vacuum solutions with either smooth or compact boundary ∂N, (c.f. §5), provided the sequence (M, g_{i}) satisfies a uniform Sobolev inequality, as for instance in (3.97), and M is irreducible, i.e. every 2-sphere in M bounds a 3-ball).

Under these conditions, Theorem A immediately implies that one has the following strong estimate

ρ(x) ≥ c(\frac{|u(x)|}{sup |u|}),  (4.23)

whenever u(x) ≠ 0. If further (4.8) holds, then Proposition 4.2 implies that (4.23) can be strengthened to a bound ρ(x) ≥ c > 0 everywhere on (M, g_{i}), i.e. there is no degeneration.

5. Completeness of the Blow-up Limits.

Theorem 3.10 proves under certain conditions the existence of non-flat blow-up limits for the sequence {g_{i}} which model the degeneration of {g_{i}} in a neighborhood of the 0-level of u. In particular, these blow-ups (D, g′) are non-trivial solutions to the static vacuum Einstein equations. However, such solutions have only been defined locally. We now address the issue of the completeness of the solutions, together with some aspects of the behavior of the potential function ū at the boundary and the renormalizations of {u_{i}} used in the construction of ū.

We first discuss the issue of completeness. Recall from Theorem 3.10 that the metric g′ is a limit of rescalings of the metrics g_{i}, i.e. g′ = lim g′_{i} based at y_{i}, where

g′_{i} = ρ(y_{i})^{−2} · g_{i},  (5.1)

Further, u_{i} is renormalized to ū_{i} satisfying

ū_{i}(y_{i}) = 1,  (5.2)

c.f. (3.69). For a fixed μ > 0 small, let

N_{i}(μ) = {x_{i} ∈ (M, g′_{i}) : ρ′(x_{i}) ≥ μ · ρ′(y_{i}) = μ},  (5.3)

where ρ′ is the L^{2} curvature radius w.r.t. g′_{i}. Let N_{i}(μ, y_{i}) be the component of N_{i}(μ) containing the base point y_{i}. By Theorem 1.5 and the non-collapse assumption (3.43), as previously discussed in §3.4, the pointed sequence of manifolds {(N_{i}(μ, y_{i}), g′_{i}, y_{i})} has a subsequence converging, weakly in L^{2,2} on compact subsets, to a non-flat connected limit domain (N(μ, y′), g′, y′). The convergence is in the strong L^{2,2} topology away from the locus {|ū_{i}| ≤ ε}, ε small, and away from the metric boundary of N_{i}(μ, y_{i}), i.e. at any given small distance away from these sets. This limit clearly extends the limit (D, g′, y′) given by Theorem 3.10. The L^{2,2} limit metric g′ on N(μ, y′) induces a natural embedding of any domain K with smooth and compact closure in N(μ, y′) into M, c.f. §1.4.

Now choose a sequence μ = μ_{j} → 0 and consider the double sequence N_{i}(μ_{j}, y_{i}). There is a diagonal subsequence {(i, j_{i})} of the double sequence {(i, j)}, with j_{i} → 0 sufficiently slowly as i → ∞, so that {N_{i}(μ_{j_{i}}, y_{i}), g′_{i}, y_{i}} converges, uniformly on compact subsets in the weak L^{2,2} topology, to a limit (N, g′, y′). Although the limit may apriori have more than one component, (the connected domains N_{i}(μ_{j_{i}}, y_{i}) may pinch off into several limit components), we assume that (N, g′, y′) is the component of the limit containing the base point y′. The limit N is contained in the union of the inclusions N(μ_{k}, y′) ⊂ N(μ_{k+1}, y′), for a sequence μ_{k} → 0, and is an open Riemannian manifold with L^{2,2} metric g′. Any smooth open domain K, with smooth boundary, and properly contained in the connected limit N, is embedded as an open domain in M via the metric g′. It is clear from thw work in §3 that the limit functions ū_{i} converge, uniformly on compact subsets, to a non-constant limit potential function ū on N.

We call (N, g′, y′, ū) the maximal solution of the static vacuum equations associated to the buffered sequence {y_{i}}. Observe that this maximal solution determined by the (convergent) sequence {y_{i}} is unique, (up to isometry), since any maximal limit must contain the initial domain (D, g′) and static vacuum solutions have unique extensions. Hence each domain N_{i}(μ, y_{i}) converges in the pointed Hausdorff topology, (c.f. [G, Ch.5A]), to its limit domain in N.

Let ∂N denote the metric boundary of (N, g′), i.e. the set of (ideal) limits of g′-Cauchy sequences, non-convergent in N. The boundary ∂N might be empty, for instance perhaps if N is the complete (isometrically doubled, c.f. Remark 5.3(ii)) Schwarzschild metric (0.17), but in ‘most’ cases one expects ∂N to be non-empty. In fact one expects ∂N to often be singular in the sense that the Riemannian metric g′ does not extend to ∂N. Of course the completion N̄ = N ∪ ∂N is a complete metric space w.r.t. the length metric induced by g′.

In case ∂N ≠ ∅, we need to describe ∂N also as a set of (ideal) limits of points of {(M, g′_{i})}. For p ∈ N̄ = N ∪ ∂N, and p_{i} ∈ M, define

p_{i} → p,  (5.4)

if, for all δ > 0, there exists q = q_{δ} ∈ N, μ = μ_{δ} > 0 and a sequence q_{i} ∈ (N_{i}(μ, y_{i}), g′_{i}) with dist_{g′_{i}}(p_{i}, q_{i}) ≤ δ, such that q_{i} → q, in the L^{2,2} convergence of N_{i}(μ, y_{i}) → N(μ, y′) discussed above. It follows from the convergence and the definition of metric completion that dist_{g′}(p, q_{δ}) ≤ δ. In particular, since δ is arbitrarily small, p in (5.4) is uniquely determined, in that there do not exist two distinct points p and p′ to which the sequence p_{i} converges.

A sequence of subsets Z_{i} ⊂ (M, g′_{i}) is said to converge in the Hausdorff topology based at y_{i} to a set Z ⊂ N̄ = N ∪ ∂N if, for each z ∈ Z, there exist z_{i} ∈ Z_{i} such that z_{i} → z in the sense of (5.4), and conversely any bounded sequence of points z_{i} ∈ Z_{i}, (i.e. dist_{g′_{i}}(z_{i}, y_{i}) ≤ C for some C), has a subsequence converging to a limit z ∈ Z in the sense of (5.4). This definition extends the notion of Hausdorff convergence of the domains N(μ, y_{i}) above to the boundary ∂N.

Observe that ∂N is formed from base points of higher order curvature concentration than {y_{i}}. Thus, p ∈ ∂N if and only if there is a sequence p_{i} ∈ (M, g_{i}), with p_{i} → p in the sense of (5.4) with ρ′(p_{i}) → 0, or equivalently, ρ(p_{i}) << ρ(y_{i}).

The next result shows that, in a certain sense, N is defined and complete at least up to the Hausdorff limits of the ε-levels L^{ε} of ū_{i}.

Theorem 5.1 (Completeness). Let (N, g′, y′, ū) be the maximal solution of the static vacuum Einstein equations associated to the (ρ, c) buffered sequence {y_{i}} in (5.1); (hence the non-collapse assumption (3.43) is assumed). Let Ū_{i}^{ε} be the component of {x_{i} ∈ (M, g′_{i}) : ū(x_{i}) > ε} containing the base point y_{i}, for ū_{i} normalized as in (5.2).

Let N^{o} ⊂ N be the maximal domain on which the potential function ū > 0. Then there is a sequence ε_{i} → 0 such that N^{o} is contained in the y_{i}-based Hausdorff limit of the domains Ū_{i}^{ε_{i}} ⊂ (M, g_{i}). Further the metric boundary ∂N^{o} ≡ L̄^{o} of N^{o} w.r.t. g′ is contained in the y_{i}-based Hausdorff limit of the ε_{i}-levels L̄_{i}^{ε_{i}} ≡ ∂Ū_{i}^{ε_{i}} of ū_{i}. The union N̄^{o} ≡ N^{o} ∪ ∂N^{o} ⊂ N̄ is complete w.r.t. the metric ḡ but may be singular at the boundary.

Let t(x) = dist_{g′}(x, ∂N^{o}), for x ∈ N^{o}. Then, as in (3.16), there is an absolute constant K < ∞ such that

|z|(x) ≤ \frac{K}{t^{2}(x)}, ū^{−1}|∇ū|(x) ≤ \frac{K}{t(x)}  (5.5)

Proof. In the notation above, consider any sequence of points {z_{i}} ∈ Ū_{i}^{ε} with dist_{g′_{i}}(z_{i}, y_{i}) ≤ R, for some arbitrary (large) constant R < ∞. Suppose further that

dist_{g′_{i}}(z_{i}, L̄_{i}^{ε}) ≥ δ,  (5.6)

for an arbitrary (small) constant δ > 0, where L̄_{i}^{ε} = ∂Ū_{i}^{ε}.

It follows from Theorem 3.3 that there is a constant a = a(ε) > 0 such that

ρ′(z_{i}) ≥ a · δ,  (5.7)

Thus, the curvature of {g′_{i}} does not blow-up (in L^{2}) within bounded distance to {y_{i}}, provided one stays a fixed distance away from the level set L̄_{i}^{ε} in Ū_{i}^{ε}. Note also that the assumption (3.43) prevents collapse in these balls B′_{y_{i}}(ρ′(z_{i})).

Let N′_{i} = N′_{i}(ε, δ) be the connected component of the set {z_{i} ∈ (M, g′_{i}) : z_{i} ∈ Ū_{i}^{ε}, dist(z_{i}, L̄_{i}^{ε}) ≥ δ} containing the base point y_{i}. Using Theorem 1.5 and Theorem 3.4 in the usual way, the pointed Riemannian manifolds {(N′_{i}, g′_{i}, y_{i})} (sub)-converge in the strong L^{2,2} topology based at {y_{i}} to a limit smooth metric g′, based at y′, defined on an open domain N′(ε, δ); the convergence is uniform on compact subsets. The functions ū_{i} converge to a limit harmonic function ū on N′(ε, δ), ū ≥ ε, so that the pair (g′, ū) is a non-flat solution of the static vacuum Einstein equations on N′(ε, δ). Of course y′ ∈ N′(ε, δ) for any ε, δ small and ū(y′) = 1. Given sequences ε_{j} → 0 and δ_{j} → 0, there exist suitable diagonal subsequences j = j_{i} of the double sequence N′_{i}(ε_{j}, δ_{j}), (with ε_{j_{i}} and δ_{j_{i}} → 0 sufficiently slowly as i → ∞), which converge, uniformly on compact subsets, to a limit. As above in the construction of N, we consider only the component (N^{o}, g′) of the limit containing y′. The connected limit (N^{o}, g′, y′) is an is an open Riemannian manifold, and is contained in the union of the domains N′(ε_{k}, δ_{k}) ⊂ N′(ε_{k+1}, δ_{k+1}) for some sequence k → 0. The limit potential function ū on N^{o} satisfies ū ≥ 0 on N^{o}, and ū(y′) = 1. Thus, ū > 0 by the minimum principle for harmonic functions. As discussed in §1.3, (N^{o}, g′) is hence a smooth solution of the static vacuum equations.

The construction of N and N^{o} implies that

(N^{o}, g′) ⊂ (N, g′),  (5.8)

(for any choice of subsequence above) and N^{o} is unique. It is possible, although not necessary, that N^{o} = N. It follows immediately from the definition preceding Theorem 5.1, (c.f. also the argument below), that (N^{o}, g′) is contained in the y_{i}-based Hausdorff limit of the domains (Ū^{ε_{i}}, g′_{i}), where ε_{i} = ε_{j_{i}} is defined above.

Let ∂N^{o} be the metric boundary of N^{o} w.r.t. g′ and let N̄^{o} = N^{o} ∪ ∂N^{o} be the metric completion. Of course ∂N^{o} ⊂ N̄.

We claim that ∂N^{o} is contained in the y_{i} based Hausdorff limit of the levels L̄^{ε_{i}}, ε_{i} = ε_{j_{i}}. To see this, suppose p ∈ ∂N^{o}, so that p = lim p_{j}, for {p_{j}} a Cauchy sequence in N^{o}. Thus, p_{j} ∈ N′(ε_{j}, δ_{j}), for some ε_{j}, δ_{j} > 0. But N′(ε_{j}, δ_{j}) = lim_{i→∞} N′_{i}(ε_{j}, δ_{j}) and hence for any j, there exist sequences q_{i,j} ∈ N′_{i}(ε_{j}, δ_{j}) with q_{i,j} → p_{j} as i → ∞, as required.

Suppose first N̄^{o} ⊂ N, (so that in particular ∂N^{o} ∩ ∂N = ∅). Then the metric g′ extends as an L^{2,2} metric past N̄^{o} and the convergence g′_{i} → g′ is in the weak L^{2,2} topology in an open region R satisfying N̄^{o} ⊂ R ⊂ N(μ, y′) ⊂ N, for some μ > 0. (More precisely, μ may depend also on the distance to the base point y′ if ∂N^{o} is non-compact). Since the functions ū_{i} also converge in L^{2,2} in R, (in fact strongly in L^{2,2} by (3.35)), it follows that ∂N^{o} is identified with the 0-level set of ū, i.e. with the event horizon Σ of (N, g′).

In this case, since g′ is L^{2,2} in a neighborhood of the event horizon Σ, as noted in §1.3, (c.f. the discussion following Theorem 1.1), elliptic regularity arguments show that in fact g′ is C^{∞} smooth across Σ and Σ is a collection of smooth totally geodesic surfaces forming the (smooth) topological boundary of N^{o}. Regardless of this fact, the weak maximum principle for L^{2,2} harmonic functions, c.f. [GT, Thm. 8.1], implies that ū < 0 in N \ N̄^{o}.

More generally, since the discussion above is local, it holds for the parts of ∂N^{o} which are contained in N, i.e. those points of ∂N^{o} which admit an open neighborhood contained in N. Now both ∂N and ∂N^{o} are closed, so the remaining points of ∂N^{o} are those contained in ∂N. Thus, the domain N^{o} is the maximal domain in N on which ū > 0. The set Z = ∂N^{o} ∩ ∂N need not be empty, and one would frequently expect ∂N^{o} = ∂N.

To conclude, the estimates (5.5) follow immediately from Theorem 3.2(II), (applied to any smooth subdomain in N^{o}), and the fact that ū > 0 on N^{o}, c.f. also (A.22) and the discussion following it at the end of the Appendix.

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Note that the limit function satisfies ū(y′) = 1, since ū(y_{i}) = 1 and, since ρ′(y_{i}) = 1, the convergence to the limit is controlled near y_{i}. In particular, ∂N does not intersect B_{y′}(1) and if ∂N^{o} intersects B_{y′}(1), it does so smoothly, as noted above.

However, it is apriori possible that all levels L̄_{i}^{ε} approach for instance the level L̄_{i}^{1} as i → ∞ away from B′_{y_{i}}(1). Thus, although the domains N′_{i}(ε, δ) contain N′_{i}(1, δ) for ε < 1, they may apriori give rise to the same limit domains, outside some region containing B_{y′}(1). In other words, the functions ū_{i} might descend from the value 1 (for example) to a value arbitrarily near 0 in arbitrarily short g′_{i} distances as i → ∞. Of course, this can only occur at points where ρ′ → 0, i.e. at ∂N. For the same reasons, we do not assert that ū(x_{j}) approaches 0 whenever x_{j} ∈ N^{o} approaches ∂N^{o} ≡ L̄^{o}.

Thus, L̄^{o} cannot necessarily be identified as the 0-level set of the potential ū, i.e. with the event horizon. Further remarks on the structure of the maximal limits (N^{o}, g′) and (N, g′) follow below.

Remark 5.2. We conjecture that for the blow-up limits (N, g′, y′) given by Theorem 5.1, one has

∂N^{o} = {ū = 0} = Σ,  (5.9)

so that ∂N^{o} is identified with the event horizon of the static vacuum solution. This amounts to proving that ū(x_{k}) → 0 whenever x_{k} converges to a point in ∂N^{o}.

Remark 5.3. (i). In general relativity, it is usually assumed that a static vacuum solution is complete up to the event horizon Σ = {ū = 0}, and in particular that ū(x) → 0 as x → Σ. Such static vacuum solutions are the most natural physically. As noted in §1.3, the static vacuum equations are formally degenerate at the locus Σ, but are formally non-degenerate away from Σ. By Theorem 3.2(I), there are no non-trivial complete solutions with empty Σ.

There are however examples of static vacuum solutions which are not complete (or defined) up to Σ, c.f. [An4, §2]. For instance, let ν be a bounded harmonic function on ℝ^{3} \ B(1), with axially symmetric, but discontinuous or non-smooth boundary values on ∂B(1). Then ν is axially symmetric and hence defines a Weyl solution as in (1.17), which does not extend everywhere past B(1). For suitable boundary values, the solution will not extend anywhere into B(1), and hence one has a solution complete away from B(1), but not defined up to the event horizon, (where ν = −∞).

Note that such examples may have max ν, and so max u, occurring on ∂B(1), while min ν occurs at infinity. In this case, such solutions formally have ‘negative mass’, c.f. §1.3.

Of course, from Remark 5.2, we conjecture that such solutions cannot arise as blow-up limits of unit volume Yamabe metrics.

(In the opposite direction, there are also solutions which are defined and smooth up to Σ, but which are not complete away from Σ, for example the A2 or B2 solutions, c.f. [EK, §2-3.6]).

(ii). While there are many static vacuum solutions which are singular at Σ, but smooth and complete away from Σ, the Schwarzschild metric (0.17) is of course smooth up to and at Σ. Here Σ is given by a round, totally geodesic 2-sphere S^{2}. Thus, the Schwarzschild metric g_{s} may be isometrically doubled across Σ giving a complete, smooth metric on S^{2} ×ℝ, asymptotically flat at both ends. This is clearly the maximal (abstract) smooth extension of g_{s}, call it Ñ. However, it is not necessarily the case that Ñ agrees with the maximal extension N defined preceding Theorem 5.1. Apriori, it is possible that the curvature of g′_{i} blows up near the event horizon Σ, (where one loses strong convergence), even though the limit in this particular case is smooth across Σ.

Basically because of this, we will usually consider only the behavior of limits up to the event horizon, (in case this is defined), i.e. within the domain N^{o} ⊂ N; c.f. however §6.5.

(iii). Although stated only for buffered sequences {y_{i}} and the associated limit, it is clear from the discussion preceding Theorem 5.1 and its proof that Theorem 5.1 remains valid for arbitrary maximal limits (N, g′, x), g′ = lim g′_{i}, g′_{i} = ρ(x_{i})^{−2} · g_{i}, for which ρ(x_{i}) → 0 and say (u_{i}/T_{i})(x_{i}) ≥ u_{o}, for some arbitrary constant u_{o} > 0. Of course in this case, the limits may be flat solutions to the static vacuum equations.

The following result, valid for general static vacuum solutions, shows that the estimate (5.5) can be improved in certain regions, c.f. also (A.27).

Lemma 5.4. Suppose (N, g, u) is an (arbitrary) static vacuum solution, u > 0 on N, and u is bounded above. Let {x_{j}} be a maximizing sequence for u in N, i.e. u(x_{j}) → sup u < ∞, and t(x) = dist(x, ∂N), where ∂N is the metric boundary of N. Then

|z|(x_{j}) ≤ \frac{μ_{j}}{t^{2}(x_{j})}, |∇logu|(x_{j}) ≤ \frac{μ_{j}}{t(x_{j})},  (5.10)

where μ_{j} = μ_{j}(x_{j}) → 0 as j → ∞.

Proof. Consider the natural rescalings of (N, g) based at {x_{j}}, i.e. the sequence of metrics (N, g_{j}, x_{j}), where g_{j} = t(x_{j})^{−2} · g_{o}. Let t_{j} = t/t(x_{j}) be the distance to ∂N w.r.t. the metric g_{j}, so that t_{j}(x_{j}) = 1. Now of course the curvature of {g_{j}} will blow up at any fixed base point x_{o} ∈ N, (with z(x_{o}) ≠ 0), if t(x_{j}) → ∞, and hence there is no limit in this region. Similarly, for example the curvature of g_{j} may blow up when t(x_{j}) → 0.

However, given any δ > 0, the curvature, (w.r.t. g_{j}), of the domain N_{δ}(j) = {p_{j} ∈ (N, g_{j}) : t_{j}(p_{j}) ≥ δ}, containing x_{j} remains uniformly bounded by the (scale-invariant) estimate (3.16). If the sequence {(N_{δ}(j), g_{j}, x_{j})} is non-collapsing at x_{j}, it follows from the (local) Cheeger-Gromov theory, (or Theorem 1.5), that a subsequence of \{B_{x_{j}}(\frac{1}{2}), g_{j}, x_{j})\} converges to a limit (B_{∞}, g_{∞}, x_{∞}). By Lemma A.2, the convergence is smooth and uniform on compact subsets, and the limit is again a smooth solution of the static vacuum equations. If the sequence instead collapses at x_{j}, then the collapse may be unwrapped by passing to the universal cover (B̃_{x_{j}}(\frac{1}{2}), g_{j}); this sequence no longer collapses, and hence sub-converges to a limit as above. (We refer to the Appendix - Lemma A.2 and Corollary A.3 - for more details of this type of standard argument).

On the other hand, it is clear that u(x_{∞}) = sup u, and so by the maximum principle, the limit harmonic function u is constant on B_{∞} and g_{∞} is hence flat. Since the convergence of N_{δ}(j) to the limit is smooth, the curvature of g_{j} is almost 0, and u is almost constant, in N_{δ}(j), away from the boundary. This implies (5.10) by scale-invariance.

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Next, we prove that the domain N^{o} ⊂ N is large in a natural sense; it particular it is unbounded.

Proposition 5.5. Let (N, g′, y′) be a maximal non-flat limit solution as in Theorem 5.1, with domain N^{o} ⊂ N on which ū > 0. There is a smooth curve γ : ℝ^{+} → N^{o}, parametrized by arclength, with γ(0) = y′, and positive constants δ_{1}, δ_{2} such that

dist(γ(s), y′) ≥ δ_{1} · dist(γ(s), ∂N^{o}),  (5.11)

and such that the cone V = V (d_{2}) = {x : dist (x, γ(s)) ≤ δ_{2} · s} over γ satisfies

V ⊂ N^{o}, i.e. V ∩ ∂N^{o} = ∅.  (5.12)

In particular, the function t(x) = dist_{g′}(x, ∂N^{o}) has linear, and hence unbounded, growth in V.

Proof. This follows from the descent construction of the limit N in Theorem 3.10. Thus, recall that the buffered sequence y_{i} = x_{i}^{k_{o}} in (M, g′_{i}), k_{o} = k_{o}(i) → ∞, as i → ∞, has predecessors x_{i}^{j′}, j′ = j′(i, j) = k_{o} − j, for any fixed j′ > 0. Thus, we have relabeled so that y_{i} = x_{i}^{j′}, for j′ = 0. By the construction, these have ū-values at least (\frac{1}{2})^{j′}, c.f. (3.73). In particular, these points are in N′_{i}(ε, 1), for ε sufficiently small, (depending on j′), c.f. the proof of Theorem 5.1; compare also with Remark 3.15(iv). Further, in the scale g′_{i} associated to y_{i} these points have very large curvature radius; in fact ρ′(x_{i}^{j′}) ≥ (d_{1})^{−j′}, for all j > 0, while both dist_{g′_{i}}(x_{i}^{j′}, y_{i}) and dist_{g′_{i}}(x_{i}^{j′}, L̄^{ε}) are also on the order of (d_{1})^{j′}, (for ε sufficiently small, depending on j′, c.f. (3.66)-(3.67)). Note that from (1.27), for any q ∈ B_{x_{i}^{j′}} (ρ′(x_{i}^{j′})), one has ρ′(q) ≥ dist_{g′_{i}}(q, ∂B_{x_{i}^{j′}} (ρ′(x_{i}^{j′}))).

Thus, let γ(s) be a path joining the limit points x^{j′} = lim x_{i}^{j′} in N^{o}, approximating a minimizing geodesic joining each x^{j′+1} to x^{j′} within B^{j′} = B_{x_{j′}} (ρ′(x^{j′})). One may then define

V = ⋃_{j′≥0} B_{x^{j′}} (ρ′(x^{j′})) ⊂ N^{o},

and the result follows.

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The curvature estimate (5.5) on N^{o} can be improved in the region V in (5.11), since the curvature radius has linear growth in V, and the predecessors x_{i}^{j′} are not (ρ, c) buffered. Thus, in fact

|z|(x) ≤ \frac{κ}{t^{2}(x)}, ū^{−1}|∇ū|(x) ≤ \frac{κ}{t(x)}  (5.13)

for x ∈ V, where κ = κ(c) may be made small by choosing the buffer constant c sufficiently small. This follows since (5.13) holds in an L^{2} sense, from the fact that x is not (ρ, c) buffered, together with elliptic regularity for the static vacuum equations, which gives an L^{∞} bound in terms of an L^{2} bound.

It seems possible that Proposition 5.5 may not hold for arbitrary ‘complete’ static vacuum solutions. (For example, consider the positive measure μ formed by placing positive multiples of the Dirac measure at all integer lattice points in ℝ^{3}, weighted so that the total mass is finite. Let v be the Newtonian potential of μ. Then there may possibly be static vacuum solutions whose potential u resembles the geometry of v).

Remark 5.6. We conclude with some remarks on the issue of the renormalization of {u_{i}} and the relation of ū with the initial sequence {u_{i}}.

Recall that the static vacuum solution constructed in Theorem 3.10 may well live in the region of (M, g′_{i}) where u_{i} is converging uniformly to 0; the potential function ū of the static limit is obtained by renormalizing, (possibly infinitely many times), the original sequence {u_{i}}. This will occur for instance if u_{i} behaved as the function v_{i} = t^{δ_{i}}, with δ_{i} → 0 as i → ∞, and t(x) = dist(x, L^{o}). Namely, if x_{ε} satisfies v_{i}(x_{ε}) = ε for any fixed ε > 0, and the distance between L^{ε} and L^{o} is scaled to size 1, then v_{i} approaches the constant function ε uniformly near x_{ε} as δ_{i} → 0.

In fact, we point out that there are static vacuum solutions (N, g, u) and points x_{j} ∈ N with t(x_{j}) = 1, (t(x) = distance to event horizon), such that

u(z_{j}) ≥ t^{δ_{j}}(z_{j}),  (5.14)

for a given sequence δ_{j} → 0, for all points z_{j} ∈ B_{x_{j}}(2). Namely, consider for example the Weyl solution (1.17) with potential function ν determined by the Riesz measure

dμ_{ζ} = \frac{1}{1 + |ζ|} dL,  (5.15)

where dL is Lebesgue measure on the axis A, and ζ is a parameter for A, c.f. (1.18). This solution is complete up to the event horizon, so L̄^{o} = Σ = {u = 0}; the event horizon corresponds to the full axis A. If x_{j} diverges to infinity in (N, g, u) at any fixed distance to Σ, then the potential function u = e^{ν} satisfies u(x_{j}) → 1 = sup u and has the property (5.14).

Thus, in addition to the discussion in Remark 5.9, this shows that the infinite descent down the u-levels in Theorem 3.10 is necessary.

It is worth noting that in this example, Σ is highly singular, and the Riesz measure dμ converges to 0 at infinity on the axis A ∼ Σ. In fact, any Weyl solution whose Riesz measure has regions of arbitrarily long length on A, with arbitrarily small, but non-zero, density w.r.t. Lebesgue measure on A will have behavior similar to (5.14).

On the other hand, although one must carry out the inductive construction in Theorem 3.10 arbitrarily many times, i.e. k_{o}(i) → ∞, as i → ∞, this does not necessarily imply an infinite descent down the levels L^{k} of u. Recall that u(x_{i}^{k}) ≥ 2^{−k}, and it may well happen that that in fact u(x_{i}^{k}) ≥ 2^{−1}, for all k ≤ k_{o}, (if c is chosen sufficiently small). This is seen specifically for example in the Schwarzschild metric, as in Example 3.9. Thus, choose initial base points {x_{i}^{1}} going to infinity in the Schwarzschild metric, so that u(x_{i}^{1}) → 1. If one carries out the inductive process of Theorem 3.10, then it is necessary to take k_{o}(i) → ∞ in order to obtain a (ρ, c) buffered sequence {y_{i}} from the sequence {x_{i}^{1}}. However, one easily sees that u(x_{i}^{k}) ≥ 2^{−1} for all k ≤ k_{o}, so that one has u(y_{i}) ≥ 2^{−1} also, (for a suitable choice of c). In this example, the inductive process in the proof of Theorem 3.10 just recaptures the original Schwarzschild metric, (up to a bounded scale change).

Similar behavior holds for Weyl solutions whose Riesz measure on A has density w.r.t. the Lebesgue measure on A either 0 or uniformly bounded below.

6. Construction of Yamabe Sequences with Singular Limits.

In this section, we discuss in detail some constructions of sequences of Yamabe metrics which illustrate the sharpness of the results in §3. In particular, these constructions exhibit the possibility that blow ups of Yamabe metrics may give rise only to trivial (i.e. flat), or super-trivial solutions of the static vacuum equations, see (3.15). (Discussions with R. Hamilton and R. Schoen were helpful to me in clarifying some aspects of the construction in Example 1). In §6.3, we analyse the behavior of the splittings (2.6) and (2.10) of z and g respectively on the (singular) limits of these sequences.

On the other hand, in §6.4 and §6.5, we construct examples of sequences of Yamabe metrics which do satisfy all the hypotheses of Theorem A(II)/Theorem 3.10, (so that this result is non-vacuous).

The existence and basic properties of these four classes of examples are explained from a somewhat more general perspective in §7.

§6.1. Example 1. Let (M, g_{o}) be a hyperbolic 3-manifold. Clearly, g_{o} is a critical point of v^{−1/3} · S on 𝕄, and thus of v^{2/3} · s on C. In fact, g_{o} is a local maximum of v^{2/3} · s on C, c.f. [Bes, 4.60].

Let N be any closed, oriented 3-manifold with σ(N) > 0, for instance N = S^{3}, S^{3}/Γ, S^{2} × S^{1}, or a connected sum of such manifolds. In the following, we will construct Yamabe metrics on the manifold N#M, which are geometrically close to the original hyperbolic manifold (M, g_{o}).

Let γ be any smooth metric (not necessarily Yamabe) of positive scalar curvature on N. Such a metric admits a positive Green’s function G_{y}(x), with pole at y, for the conformal Laplace operator −8Δ_{γ} + s_{γ}, c.f. [LP, Thm2.8]. Consider the conformally related metric

g = G_{y}^{4}(x) · γ,  (6.1)

for any fixed y ∈ N. This metric is a complete, scalar-flat metric on N \ {y}, which is asymptotically flat in the sense that, outside a large compact set, (thus in a small neighborhood of {y}),

g_{ij} = (1 + \frac{2m}{r}) δ_{ij} + O(r^{−2}),  (6.2)

c.f. (1.15). Here m is the mass of the metric satisfying m ≥ 0, with equality if and only if g is flat, by the positive mass theorem [SY]. From [Sc1], the metric g is flat only when (N, γ) is the canonical constant curvature metric on S^{3}. Note that the curvature tensor R_{g} of g satisfies the decay condition

|R_{g}|(x) ≤ c · r(x)^{−3},  (6.3)

as r(x) → ∞.

Choose any fixed value of ε, with 0 < 4ε < i_{o} ≡ inj_{g_{o}}(M), where inj denotes the injectivity radius. Given any x ∈ M, we may then glue in the metric g above to B_{x}(2ε) ⊂ (M, g_{o}) as follows. Given a fixed center point p ∈ N \ {y}, let B(R) denote the geodesic R-ball in (N \ {y}, g) centered at p. Scale g|_{B(R)} to size ε, i.e. define

g_{ε} = (\frac{ε}{R})^{2} · g,  (6.4)

so that g_{ε}|_{B_{x}(ε)} is homothetic to g|_{B_{p}(R)}. From (6.2) and (6.3), it follows that in the geodesic annulus A(ε, 2ε), the sectional curvature of g_{ε} is on the order of |K_{ε}| = O(ε^{−2}R^{−1}). Hence the curvature is bounded in A(ε, 2ε) if ε^{2}R is bounded away from 0, while the metric g_{ε} is almost flat in this band if ε^{2}R >> 1. For simplicity, we assume from now on that ε^{2}R >> 1. Note that the unit ball B_{p}(1) in (N, g) is of radius ε/R in g_{ε}; for ε^{2}R > 1, ε/R >> ε^{3}.

In order to bend the metric g_{ε} on B(2ε) so it has almost constant scalar curvature −6 = s_{g_{o}}, define

g̃_{ε} = ψ^{4} · g_{ε},  (6.5)

where ψ = 2^{1/2}(1 − τ^{2})^{−1/2} on B(2ε) and where τ will be determined below. Note that if g_{ε} were flat on B(2ε) and τ = t = dist_{g_{ε}}(x, ·), the metric g̃_{ε} is the hyperbolic metric of scalar curvature −6. Since g_{ε} is scalar-flat a simple computation using (1.12) shows that

s̃_{ε} = −(1 − τ^{2})Δτ^{2} − 6τ^{2}|dτ|^{2}.  (6.6)

We choose τ so that Δτ^{2} = 6, and τ is close to the distance function t(z) = dist_{g_{ε}}(z, x). Hence (6.6) becomes

s̃_{ε} = −6[1 + τ^{2}(|dτ|^{2} − 1)].  (6.7)

To construct such functions, return to the asymptotically flat manifold (N \ {y}, g). We claim there is a function α on N, with Δα = 6 and α asymptotic to r^{2}. To define α consider the function r^{2} + mr + 2mlnr; this is defined and smooth outside a compact set K of (N \ {y}, g), and off K, one computes that Δ(r^{2} + mr + 2mlnr) = 6 + 4mr^{−3} + 0(r^{−4}). Let h be a smooth extension of r^{2} + mr + 2mlnr to all of N \ {y}; then Δh = f, where the function f satisfies |6 − f| = 0(r^{−3}). We define α as α = h + ϕ, where ϕ(q) = − ∫ G(q, z)(6 − f)dz and G is the (positive) Green’s function for Δ on N \ {y}. Since the product dist(q, z) · G(q, z) is bounded below and above, ϕ is well defined and decays at infinity as 0(r^{−1}). It follows that the function α satisfies Δα = Δ(h + ϕ) = 6 on (N \ {y}, g). By adding a suitable constant, we may assume α > 0.

Rescaling to the metric g_{ε}, we then define τ = (ε/R)α^{1/2} on B_{x}(2ε). Since the metrics g and g_{ε} are homothetic, scaling properties imply Δ_{g_{ε}}τ^{2} = 6 and |dτ|_{g_{ε}} = |dα^{1/2}|_{g} ≈ 1 in A(ε, 2ε), while |dτ|_{g_{ε}} is bounded everywhere. Note also that τ → 0 uniformly in B_{x}(2ε) as ε → 0.

It follows from (6.7) that the metric g̃_{ε} has scalar curvature converging to −6 in B(2ε) as ε → 0 and ε^{2}R → ∞. Further, from the remarks following (6.5) the metric g̃_{ε} approaches the hyperbolic metric in A(ε, 2ε) in the C^{2} topology, as ε → 0 with ε^{2}R → ∞. Thus, the metric g̃_{ε} may be perturbed a small amount in C^{2} in A(ε, 2ε) to match with the hyperbolic metric g_{o} at S(2ε) to give a smooth metric, again called g̃_{ε} on N#M, with scalar curvature close to −6 everywhere. Note that the conformal factor ψ converges pointwise to the conformal factor ψ_{o} bending the flat metric to the hyperbolic metric. In particular, on (B_{x}(2ε), g_{ε}), we have

ψ = ψ_{ε} → 2^{1/2}, as ε → 0, ε^{2}R → ∞.  (6.8)

Finally, let ḡ_{ε} be the Yamabe metric in the conformal class of g̃_{ε} with the same volume. Thus

ḡ_{ε} = w^{4} · g̃_{ε},  (6.9)

where w satisfies

w^{5}s̄_{ε} = −8Δw + s̃_{ε}w.  (6.10)

Noting that min w < 1 < max w and evaluating (6.10) at points realizing the minimum and maximum of w, gives the estimates

min w ≥ inf |\frac{s̃_{ε}}{s̄_{ε}} | , max w ≤ sup |\frac{s̃_{ε}}{s̄_{ε}} | .  (6.11)

Since s̃_{ε} converges to −6 in the C^{o} norm as ε → 0 and ε^{2}R → ∞, each ratio in (6.11) must converge to 1, so that

w → 1  (6.12)

in the C^{o} topology. Thus, the family of Yamabe metrics {ḡ_{ε}} on N#M have scalar curvatures converging to −6 = s_{g_{o}}, and clearly converge smoothly to the hyperbolic metric on M away from the point x. The manifold N is being crushed to the point x under {ḡ_{ε}}, as ε → 0.

It is easy to see that {ḡ_{ε}} is a degenerating family in the sense of (0.8), that is

∫_{N#M} |zḡ_{ε}|^{2} dVḡ_{ε} → ∞, as ε → 0,

provided (N, γ) ≠ (S^{3}, g_{can}). Namely, in this case, note that the metric g on N \ {y} satisfying (6.2) has a definite amount of curvature, say

0 < ∫_{B_{p}(R)} |z_{g}|^{2} dV_{g} = κ < ∞,  (6.13)

for R large. The scaling properties of curvature then imply that

∫_{N#M} |zg̃_{ε}|^{2} dVg̃_{ε} ≥ \frac{1}{2} ∫_{B(ε)} |z_{g_{ε}}|^{2} dV_{g_{ε}} = \frac{κ}{2} · \frac{R}{ε} → ∞, as ε → 0,  (6.14)

where the first inequality follows from (6.8) and elliptic regularity applied to the equation defining ψ. (The factor \frac{1}{2} can be replaced by any number < 1). Similarly, the estimate (6.14) then also holds for the Yamabe metrics {ḡ_{ε}}, using (6.12) and elliptic regularity for the equation (6.10) defining w.

Observe however that all blow-ups by the L^{2} curvature radii of {ḡ_{ε}}, as ε → 0, ε^{2}R → ∞, have either flat limits, or are exactly the limit metric g, given by (6.1) above. It is clear that this metric can only be a super-trivial solution of the static vacuum equations, with potential function u identically zero. For instance, since the metric g is asymptotically flat and smooth everywhere, Theorem 1.1 implies that any non-trivial static vacuum solution would have to be the Schwarzschild metric, which is obviously not isometric to g. Thus all blow-up limits of {ḡ_{ε}} are either trivial, i.e. flat, or are super-trivial. It is easily seen that hypothesis (i) and, using Theorem 2.10, hypothesis (ii) of Theorem A(II) are satisfied. In particular, this shows that the assumption (iii) in Theorem A(II) is necessary.

We make several further remarks on this construction, c.f. also §6.3.

Remark 6.1. (i). It is obvious that this construction can be carried out on any collection of disjoint 4ε balls in (M, g_{o}). Choosing a sequence R_{i} → ∞ and ε_{i} → 0, so that ε_{i}^{2} · R_{i} → ∞, one may construct sequences of Yamabe metrics {g_{i}} on manifolds of the form

M_{k_{i}} = (#_{1}^{k_{i}}N_{j}) #M,

where N_{j} is any closed oriented manifold with σ(N_{j}) > 0 and {k_{i}} either bounded or divergent to ∞. Such sequences satisfy v^{2/3}s(g_{i}) → v^{2/3}s(g_{o}), and may be arranged that their curvature blows up on a progressively denser set in any prescribed domain (Ω, g_{o}) ⊂ (M, g_{o}).

(ii). One may also glue in metrics of the form (6.1), where the Green’s function based at y is replaced by a finite sum of Green’s functions based at a finite number of poles {y_{i}} ∈ N. This gives rise to a complete scalar-flat metric with a finite number of asymptotically flat ends E_{i}, each of which may be glued as above into a small ball of a hyperbolic manifold M_{i}. The limit as ε → 0 is then a finite number of hyperbolic manifolds, glued together at one point.

(iii). It is easy to see that this construction does not require that the gluing be into a hyperbolic manifold (M, g_{o}). The same procedure is valid on any 3-manifold (M, g) with Yamabe metric g of scalar curvature s_{g} < 0.

Namely, it is a standard result that given any such (M, g) and x ∈ M, there is a δ = δ(M, g, x) > 0 such that the metric g may be deformed to a metric ĝ within B_{x}(δ) so that ĝ is the constant curvature metric of scalar curvature s_{g} in B_{x}(δ^{2}) and such that the scalar curvature of ĝ is arbitrarily close, (depending only on δ), to the constant s_{g}. Such deformations are local, in the sense that ĝ = g outside B_{x}(δ), and further small, in the sense that |ĝ − g|_{C^{o}} ≤ δ′ = δ′(δ) in B_{x}(δ). A proof of this fact is given for instance in [Kb1, Lemma 3.2].

As an example, for any ε > 0, this glueing construction may be applied to the manifold (N#M, ḡ_{ε}) constructed above; thus, one may choose ε_{1} << ε and glue another manifold N_{1} into a small ε_{1}-ball in (N, ḡ_{ε}) ⊂ (N#M, ḡ_{ε}) for instance, giving Yamabe metrics on N_{1}#N#M. Again, this process may be repeated inductively. Thus, one produces examples with curvature going to infinity at different scales or rates.

These variations of the construction have all blow-up limits which are either super-trivial or trivial solutions of the static vacuum equations, c.f. Remark 6.8 for further discussion; (note the one possible exception there however). Hence the geometry of blow-up limits near the locus {u = 0} can be quite arbitrary, in contrast to the geometric structure of blow-ups obtained when the hypotheses of Theorem A(II) are satisfied. The only common structure of the blow-up limits here is that they are scalar-flat and asymptotically flat.

§6.2. Example 2. Let (M, g_{o}) be as in Example 1. Here, we will construct degenerating sequences of Yamabe metrics on connected sums, whose blow-up limits are the (doubled) Schwarzschild metric.

As indicated in the discussion in §3.1, static Einstein metrics are closely related to solutions, possibly only locally defined, of the equation L^{∗}u = 0. Consider then static Einstein manifolds of the form X = Ω ×_{h} S^{1}, with scalar curvature −12. Analogous to the (scalar-flat) Schwarzschild metric, consider the metric g on Ω = ℝ^{+} × S^{2}, given by

g = dt^{2} + f^{2}(t)ds_{S^{2}}^{2},  (6.15)

where f is the solution to the following initial value problem; f(0) = a > 0, f′ ≥ 0,

(f′)^{2} = 1 + f^{2} − (a^{3} + a)f^{−1},  (6.16)

see [Bes, Ch. 9J]. The scalar curvature of g is −6. Setting h = f′, one computes that

L^{∗}h = 0,  (6.17)

and X is Einstein, with Ric_{X} = −3 · g_{X} by Proposition 3.0. The 2-sphere S = {t = 0} = {f = a} = {h = 0} is totally geodesic and of constant curvature a^{−2}, while the metric g is asymptotic to the hyperbolic metric H^{3}(−1). In fact, as a → 0, the metric g = g_{a} converges to the hyperbolic metric H^{3}(−1) off S, while S is being crushed to a point. The curvature goes to infinity in small neighborhoods of S, and if one rescales g_{a} by the L^{2} curvature radius, the blow-ups converge to the Schwarzschild metric.

Thus, as in Example 1, choose a small ball B_{x}(ε) ⊂ (M, g_{o}). The smallest choice for ε is ε ∼ a^{1/3}. For any ε ≥ a^{1/3}, the neighborhood N_{ε/2,ε} = {x : dist(x, S) ∈ (ε/2, ε)} of S in (Ω, g_{a}) has uniformly bounded curvature as a → 0, and for ε >> a^{1/3} is almost isometric to the hyperbolic annulus A(ε/2, ε) ⊂ B_{x}(ε). As in Example 1, we assume for simplicity that ε >> a^{1/3}, but ε → 0 as a → 0. Thus, by a small smooth perturbation, one may glue on N_{0,ε} = I × S^{2} with metric g_{a} onto A(ε/2, ε) to obtain a metric g_{a} on M \ (3-ball), with scalar curvature almost −6, and with totally geodesic, constant curvature boundary S = S^{2}(a).

This process may be performed on any other hyperbolic manifold (M′, g_{o}), and by matching at the isometric boundaries S, one obtains a smooth metric g_{a} on M′#M, with scalar curvature almost −6, and with g_{a} isometric to g_{o} outside the ε -neighborhood of S. The metric g_{a} may then be conformally deformed to a Yamabe metric ḡ_{a}. One proves as in (6.10)-(6.12) that ḡ_{a} is almost isometric to g_{a}; the conformal factor w converges to 1 in the C^{o} topology as a → 0.

Here, the limit of (M′#M, ḡ_{a}) as a → 0 is the 1-point union, at x, of the hyperbolic manifolds M and M′. The limit of blow-ups by the L^{2} curvature radius at x, or at points where h_{a} = \frac{1}{2} for example, is the Schwarzschild metric (0.17) doubled isometrically across the event horizon.

Consider briefly the behavior of the function h = h_{a}, as a → 0. Clearly, on any fixed interval [t_{o}, μ), for t_{o} > 0, the functions h_{a} converge smoothly to the function cosh t > 1. However, from (6.16), h_{a}(0) = 0, h′_{a}(0) = a^{−1}, so that h_{a} increases very rapidly from 0 to 1; compare with Propositions 3.18 or 3.19. In particular, the analogue of the descent construction of Theorem 3.10, with the function h_{a} in place of u_{i}, gives rise to blow-ups limits given by the Schwarzschild metric with limit potential function h = (1 − 2mt^{−1})^{1/2}, c.f. (0.17).

However, for the metrics ḡ_{a} on M′#M, there is no non-trivial descent down the u-levels, that is the function u does not satisfy the hypothesis (iii) of Theorem A(II), or Proposition 3.18/3.19. In fact, the limit function u is identically 0 on the (doubled) Schwarzschild blow-up limit, giving a super-trivial solution as in Example 1.

To see this, if u were non-zero on the Schwarzschild blow up, one must have

u = c · h,  (6.18)

for some constant c ≠ 0. (Since the limit is the Schwarzschild metric, as noted in Example 3.9 and Remark 5.6, the descent construction in Theorem 3.10 terminates at levels where u_{a} is bounded away from 0, so that the renormalization to ū_{a} is not necessary).

Now, by Proposition 2.9, Δu is uniformly bounded in L^{2} (as a → 0). Thus,

Δu → 0 in L^{2},  (6.19)

in the blow-ups, (as in the proof of Proposition 3.1). Thus, in passing from M′ into M through S, the function u must change sign; in the blow-up Schwarzschild limit, u is odd w.r.t. reflection in the core S, and is asymptotic to −c or +c at either end of the doubled Schwarzschild metric, corresponding to a region in M′ or M respectively. This implies that the functions u = u_{a} for the metrics ḡ_{a} converge to a limit function u on the hyperbolic manifolds M′ \ {x} and M \ {x} with

u(y) → −c as y → x in M′, while  (6.20)
u(y) → +c as y → x in M.

It turns out that the behavior (6.20) is never possible when c ≠ 0, c.f. Remark 6.8, although the proof is non-trivial. We prove the impossibility of (6.20) here in a simple special case, namely when M = M′. In this situation, there is an isometry ι of (M#M, ḡ_{a}) defined by reflection in the core 2-sphere S = S^{2}(a). Let u′_{a} = u_{a} ∘ ι. From the invariance of the L^{2} metric (1.3), the L^{2} orthogonality of L^{∗}u and ξ, and the invariance of ḡ_{a} under ι, it follows easily that L^{∗}(u_{a} − u′_{a}) = 0 on (M#M, ḡ_{a}). But Ker L^{∗} = 0, (c.f. the beginning of §2), so that u_{a} = u′_{a}. Thus, u_{a} is an even function w.r.t. reflection in S. This property passes to the limit, (or the blow-up limit), and hence shows (6.20) is impossible.

Remark 6.2. (i) With only minor changes, this construction can be performed more generally on any pair of manifolds M, N, with Yamabe metrics of negative and equal scalar curvatures, using the local deformation to hyperbolic metrics in Remark 6.1(iii). We point out that both the parameters ε and a must be chosen sufficiently small, (in addition to satisfying a^{1/3} << ε), depending on the choice of Yamabe metrics on M and N.

To compare with Example 1, suppose g is any Yamabe metric on a manifold M with scalar curvature −6 say. Note that any manifold N with σ(N) > 0 carries unit volume Yamabe metrics with scalar curvature −δ, for any δ > 0. Such a metric may be rescaled so that the scalar curvature is −6 , so that it then has volume on the order of δ^{3/2} and diameter on the order of δ^{1/2}. The construction in Example 2 may then be carried out on the pair M and N, (here ε will be much smaller than δ). This gives rise to a sequence of Yamabe metrics {g_{i}} on N#M, which converge smoothly on M \ {pt} to g_{o}, while crushing N to a point. Thus, this sequence has the same basic features as the sequence in Example 1, except that blow-up limits are given by the isometrically doubled Schwarzschild metric. The triviality of the limit potential function u is discussed in Remark 6.8.

(ii). In fact, the construction in Example 2 is more general in most respects than that of Example 1, since one may form in addition connected sums with all manifolds N satisfying σ(N) ≥ 0, (for example all graph manifolds). Namely, any such N has unit volume Yamabe metrics with scalar curvature −δ. These may be scaled to make the scalar curvature −6 and volume ∼ δ^{3/2} and the construction on N#M proceeds as above. When σ(N) = 0, the diameter of these rescaled metrics may remain large however, so that N may no longer be crushed to a point, but be (volume) collapsed, away from {pt}, to a possibly arbitrarily long lower dimensional space. (The volume collapse may not necessarily be with uniformly bounded curvature).

(iii). Similarly, all the constructions mentioned in Remark 6.1 can also be recaptured by modifications of the construction in Example 2, as above, except the construction in Remark 6.1(ii). Regarding this construction, we note that there are no smooth and complete static vacuum solutions which have more than 2 ends; for instance there is no complete static vacuum solution on a k-punctured 3-sphere, with k ≥ 3. The Schwarzschild metric is a complete, conformally flat and scalar-flat metric on a 2-punctured 3-sphere.

§6.3. In this subsection, we analyse in somewhat greater detail the structure of the degeneration in Examples 1 and 2, in particular the structure induced on the limit (singular) manifold.

Throughout this subsection, let

M_{o} = ⋃_{1}^{n} M_{k}  (6.21)

be a finite collection of connected closed hyperbolic manifolds, identified at a point x, with hyperbolic metric g_{o}. Let {g_{i}} be a sequence of Yamabe metrics on a closed connected manifold M, converging in the Gromov-Hausdorff topology to (M_{o}, g_{o}) and converging smoothly to the hyperbolic metric g_{o} on M_{o} \ {x}. For example, M may be the connected sum of the components M_{k}, or M may be of the form M = N#M_{o}, where M_{o} \ {x} is connected, (i.e. n = 1), and σ(N) > 0, so that the sequence {g_{i}} crushes N to the point {x}.

We also assume throughout §6.3, (until Remark 6.8 at the end), that {g_{i}} has no blow-ups which are non-trivial solutions to the static vacuum equations with potential u ≢ 0. By the various constructions in §6.1 and §6.2, such sequences exist at least for many configurations M_{o}.

From Theorem 2.10, it is clear that the L^{2} norm of z^{T} is uniformly bounded for the sequence {g_{i}}, and hence by (2.35), λ is bounded away from 0 and so δ = 1−λ is bounded away from 1. Further, from Proposition 4.2 combined with Theorem A, the function f = f_{i} does not converge to any constant in L^{2}. In particular we see that, for some subsequence,

δ = δ_{i} → δ_{o} > 0.  (6.22)

(If δ_{i} → 0, then from (2.35), z^{T} → 0 in L^{2}, so that from the proof of Proposition 4.2, it follows that u → 1 or f → 0 in L^{2}, which is impossible by the statement above).

The convergence of the metrics g_{i} to the limit g_{o} is smooth away from x. Hence for instance the trace-less Ricci curvature z_{i} converges smoothly to the limit z = 0 away from x. Since f_{i} is uniformly bounded in T^{2,2}(M, g_{i}) by Proposition 2.9, f_{i} converges at least weakly in T^{2,2} away from x to a limit function f ∈ T^{2,2}(M_{o}, g_{o}), while ξ_{i} converges weakly in L^{2} to a limit ξ ∈ L^{2}(M_{o}, g_{o}). In fact, the convergence of f_{i} and ξ_{i} to their limits is smooth away from x, as one sees by applying elliptic regularity to the equation (2.29). Since Δf ∈ L^{2}(M_{o} \ {x}, g_{o}), and each component (M_{k} \ {x}, g_{o}) of (M_{o} \ {x}, g_{o}) extends smoothly to the hyperbolic metric on M_{k}, elliptic theory implies that f is bounded in L^{2,2} and hence is a C^{1/2} function on each M_{k}, c.f. [GT, Thm.8.12].

The next result shows that f extends continuously through x on M_{o}.

Lemma 6.3. On the limit (M_{o} \ {x}, g_{o}), we have

f(y) → −1  as y → x,  (6.23)

in any component M_{k} of M_{o} \ {x}.

Proof. To see this, first note that since the curvature of {g_{i}} blows up near x, Theorem 3.3 implies that u_{i} goes to 0 somewhere near x. Since there are no non-trivial blow-up limit solutions, Proposition 3.18 and Theorem 3.10 imply that for any ε > 0, with B_{i} = (B_{x}(ε), g_{i}),

osc_{B_{i}} u_{i} ≤ δ = δ(ε),  (6.24)

where δ(ε) → 0 as ε → 0. (Note also that T_{i} is uniformly bounded by the remark following the proof of Proposition 4.2). It follows that (6.24) holds on the limit (M_{o}, g_{o}), which gives (6.23).

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The basic identity (2.6), that is

z_{i} = L^{∗}f_{i} + ξ_{i},

on (M, g_{i}), passes to the limit and gives the equation

0 = L^{∗}f + ξ,  (6.25)

on (M_{o} \ {x}, g_{o}).

Of course there are no non-trivial smooth solutions to (6.25) on all of (M_{o}, g_{o}), since the terms are L^{2} orthogonal. However, this is not the case in the presence of the singular point {x}.

Lemma 6.4. The terms L^{∗}f and ξ in (6.25) are not identically 0 on each component M_{k} \ {x} of M_{o} \ {x}, so that the equation (6.25) is non-trivial on each M_{k}.

Proof. Suppose ξ = 0 on M_{k} \ {x} for some k. The trace equation (2.12), valid for f = f_{i} on (M, g_{i}), passes to the limit away from {x}, so that we would then have

2Δf + sf = 0,

on M_{k} \ {x}. Since f is bounded, a standard (Riemann) removable singularity result implies that f extends smoothly over {x} to give a smooth solution to this equation on M_{k}. However, since s = −6 < 0, the operator −Δ − \frac{s}{2} is positive, and thus one must have f ≡ 0 on M_{k}. This contradicts Lemma 6.3.

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In particular, on each (M_{k} \ {x}, g_{o}), we have a non-trivial solution of the trace equation

2Δf + sf = tr ξ.  (6.26)

One sees easily that if ξ were bounded in a neighborhood of {x} in (M_{k} \ {x}, g_{o}), then (6.25) holds weakly across {x}, which implies that ξ vanishes; more precisely, one uses a simple cutoff function argument to prove this. Thus ξ is unbounded near {x}, as is tr ξ, on each component M_{k} \ {x}.

To analyse the behavior of tr ξ, since L(ξ) = 0, δξ = 0 and z = 0 on (M_{o} \ {x}, g_{o}), we have from the defining equation (2.2)

Δ tr ξ + \frac{s}{3} tr ξ = 0.  (6.27)

Exactly as in the proof of the non-triviality of (6.25), note that this equation has no bounded non-zero (weak) solutions on compact manifolds. Since tr ξ ∈ L^{2}(M_{o} \ {x}), and tr ξ is smooth away from x, it follows that, (up to a multiplicative constant), the unique non-zero solution to (6.27) is given by the Green’s function G_{x}(y) for the positive operator −Δ − \frac{s}{3} = −Δ + 2.

Thus, tr ξ is asymptotically of the form c_{k}/t, t = dist_{g_{o}}(·, x), as t → 0 on each M_{k}, i.e.

t · tr ξ → c_{k},  (6.28)

for some constant c_{k} ≠ 0. The next result evaluates the constant c_{k}, c.f. also Lemma 2.3.

Lemma 6.5. On each component M_{k} \ {x} of M_{o} \ {x}, we have

∫_{M_{k}} |ξ|^{2} = −\frac{s}{3} ∫_{M_{k}} tr ξ = 4πc_{k} > 0.  (6.29)

Proof. Let η = η_{k} be a cutoff function with η ≡ 1 on M_{k} \ B_{x}(ε), η ≡ 0 on B_{x}(ε/2), for ε small. Multiplying the equation (6.25) by ηξ gives

∫ η|ξ|^{2} = ∫ ⟨L^{∗}f, ηξ⟩ = ∫ fL(ηξ) = f(x) ∫ L(ηξ) + O(ε),  (6.30)

where O(ε) → 0 as ε → 0. The last equality follows from Lemma 6.3. As above in (6.27), from (2.2) and the facts that L(ξ) = 0 and δξ = 0, one computes

L(ηξ) = − tr ξΔη − 2⟨d tr ξ, dη⟩ + ⟨D^{2}η, ξ⟩.  (6.31)

Thus, the divergence theorem gives

∫ L(ηξ) = ∫ ηΔ tr ξ + ∫ ⟨dη, δξ⟩ = ∫ ηΔ tr ξ = −\frac{s}{3} ∫ η tr ξ,  (6.32)

where the last inequality follows from (6.27). Hence, combining (6.30) and (6.32) and letting ε → 0 gives

∫_{M_{k}} |ξ|^{2} = −\frac{f(x)s}{3} ∫_{M_{k}} tr ξ,

and the first equality in (6.29) follows from Lemma 6.3.

For the second equality, apply the divergence theorem to (6.27) to obtain

−\frac{s}{3} ∫_{M_{k}\B_{x}(ε)} tr ξ = ∫_{S_{x}(ε)}⟨d tr ξ, ν⟩,

where ν is the outward unit normal. Since ν = −\frac{d}{dt}, use (6.28) to obtain

lim_{ε→0} ∫_{S_{x}(ε)}⟨d tr ξ, ν⟩ = 4πc_{k}.

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The maximum principle applied to (6.27) implies that tr ξ does not change sign on any component M_{k} \ {x} of M_{o} \ {x}. Hence, from Lemma 6.4, on each M_{k},

tr ξ > 0.  (6.33)

Returning to (6.26), it follows from Lemma 6.3 and (6.28) that near x on M_{k} \ {x}, f has the expansion

f = −1 + a_{k}t + o(t), as t → 0,  (6.34)

where a_{k} = c_{k}/4 > 0. In particular, f extends to a Lipschitz function on each (M_{k}, g_{o}). Finally, from (6.25), we have

LL^{∗}f = 0,  (6.35)

on (M_{o} \ {x}, g_{o}). Since z = 0 on M_{o} \ {x}, from (2.27), (6.35) is of the form

2ΔΔf + \frac{5}{3}sΔf + \frac{s^{2}}{3} f = 0.  (6.36)

It is easy to see, again from standard removable singularity results, that there are no non-zero C^{2} solutions to (6.36) on M_{k}, so that f does not extend to a C^{2} function on M_{k}. Observe that since tr ξ > 0 by (6.33), the maximum principle applied to the trace equation (6.26) implies

f < 0 on M_{o}.  (6.37)

Recall that tr ξ is, (up to a multiplicative constant), the positive Green’s function or fundamental solution of the operator Δ + \frac{s}{3} on M_{k}, with pole at x. Analogously for f, we have:

Proposition 6.6. The function f satisfying (6.35) and (6.34) is, up to a multiplicative constant, the fundamental solution of the elliptic operator LL^{∗} on M_{k}, with singularity at {x}.

Further, the function f is the unique solution of (6.35) on M_{o}, smooth on M_{o} \{x} with |f(y)+1| = O(t) as t → 0, where t(y) = dist(y, x) in M_{o}.

Proof. The leading order term of LL^{∗} is the bi-Laplacian Δ^{2}, whose fundamental solution F_{x}(p) based at x is asymptotic to c · t(p) = c · dist(x, p), for some c ≠ 0. Since f is smooth away from {x}, this implies the first statement.

The second statement follows from the fact that the only solution of the equation LL^{∗}h = 0, with |h| = O(t), as t → 0, is h ≡ 0. To see this, standard elliptic regularity applied to the equation (6.36) away from x implies that

|Δh| · t^{2} = o(t) and |∇h| = O(t),

as t → 0. Now pair (6.36) with η · h, where η is a cutoff function as in Lemma 6.5, with supp|∇η| ⊂ A(ε, 2ε), |∇η| ≤ c/ε, |D^{2}η| ≤ c/ε^{2}. Integrating by parts gives

∫ η(Δh)^{2} + η|s||∇h|^{2} + ηs^{2}h^{2} = − ∫ Δh[hΔη + 2 < ∇h, ∇η >] + s ∫ h < ∇h, ∇η >.

The estimates above on h and its derivatives imply that the right side of this equation goes to 0, as ε → 0. Since all terms on the left are non-negative, it follows that h = 0.

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Since the function f exists on M_{o}, Proposition 6.6 implies that (conversely), independent of how f was constructed as a limit of {f_{i}} from the geometry of (M, g_{i}), on any component M_{k} of M_{o} there exists a unique solution to (6.35) or (6.36), smooth away from x, which approaches the value −1 at x linearly in t. Such a solution has the expansion (6.34), and hence the constant a_{k} = c_{k}/4 > 0 in (6.34) is an invariant of the punctured manifold (M_{k}, x). It might be considered as a kind of mass for the operator LL^{∗} at {x}. It follows that the terms L^{∗}f and ξ in (6.25) exist and are determined by the geometry of each (M_{k}, x), independent of the approximating Yamabe sequence.

The fact that ξ and f are non-zero on the limit (M_{o} \ {x}, g_{o}) shows that they detect the ‘bad’ convergence of {g_{i}} near the singular point {x}, even on the limit. If {γ_{i}} is a sequence of Yamabe metrics on a connected hyperbolic manifold M_{o} = M, converging smoothly to (M_{o}, g_{o}) everywhere, then of course ξ → 0, z^{T} → 0 and f → 0 smoothly. When M ≠ M_{o}, it is obvious though that the metric g_{o} itself extends smoothly over {x} to give the hyperbolic metric on the closed manifold M_{o}. This is related to the following consequence of Lemma 6.5.

Corollary 6.7. For (M, g_{i}) as above, we have

lim_{i→∞} ∫_{M_{k}} |ξ_{i}|^{2}dV_{g_{i}} = ∫_{M_{k}} |ξ|^{2}dV_{g_{o}}  (6.38)

and

lim_{i→∞} ∫_{M_{k}} |L^{∗}u_{i}|^{2}dV_{g_{i}} = ∫_{M_{k}} |L^{∗}u|^{2}dV_{g_{o}}.  (6.39)

Proof. Theorem 2.2, (2.10), and the continuity of the scalar curvature and volume in the convergence to the limit (M_{o}, g_{o}) show that (6.38) and (6.39) are equivalent. Fatou’s theorem, (lower semi-continuity), implies that, for each k,

∫_{M_{k}} |ξ|^{2}dV_{g_{o}} = ∫_{M_{k}\{x}} |ξ|^{2}dV_{g_{o}} ≤ lim inf_{i→∞} ∫_{M_{k}\{x}} |ξ_{i}|^{2}dV_{g_{i}},  (6.40)

where for the integral on the right we consider M_{k} \ {x} ⊂ M via a Gromov-Hausdorff approximation. Summing over k then gives

∫_{M_{o}} |ξ|^{2}dV_{g_{o}} ≤ lim inf_{i→∞} ∫_{M} |ξ_{i}|^{2}dV_{g_{i}} = lim inf_{i→∞} −\frac{s}{3} ∫_{M} tr ξ_{i} dV_{g_{i}} = −\frac{s}{3} ∫_{M_{o}} tr ξ dV_{g_{o}} = ∫_{M_{o}} |ξ|^{2}dV_{g_{o}}.

Here, the first equality uses (2.14), the second uses the fact that ξ_{i} → ξ weakly in L^{2} and the third follows from Lemma 6.5. It follows that equality holds in (6.40) and the lim inf may be replaced by limit.

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Of course the L^{2} norm of L^{∗}f_{i} and of z_{i} diverges to ∞ as i → ∞. In other words, Corollary 6.6 shows that the splitting of the metric g_{i} in (2.10) is continuous in the limit i → ∞, but the splitting of the curvature z_{i} is far from continuous.

As a curiosity at this stage, we note that an easy computation, using the relation (6.29), implies

\frac{d}{dt}(v^{2/3} · s)(g_{o} + tξ)|_{t=0} = −v^{−1/3} ∫_{M_{o}\\{x\}} |ξ|^{2}dV.  (6.41)

Thus deforming the metric g_{o} on M_{o} \ {x} in the direction −ξ increases the Yamabe constant v^{2/3} · s of g_{o}. Of course since ξ is unbounded, this is not a smooth perturbation of the hyperbolic metric on M_{o}.

Remark 6.8. In this remark, we consider briefly the converse of the previous discussion in §6.1-§6.3, namely to what extent blow-up limits of Yamabe metrics converging to a singular hyperbolic limit as above must be super-trivial solutions of the static vacuum equations.

Thus, let (M, g_{i}) be any sequence of Yamabe metrics converging to M_{o} ∪X, where M_{o} is as in (6.21) and X, (possibly empty) is a lower dimensional space of zero volume, compare with Remark 6.2(ii). Suppose further that the blow-up limit at the singular point x ∈ M_{o} is a complete, smooth, non-flat and asymptotically flat manifold (N, g_{b}), i.e. all ends are asymptotically flat. All of the constructions and their variations in §6.1 and §6.2 satisfy these assumptions, (assuming the construction is not iterated).

By Theorem A(I), the blow-up limit (N, g_{b}) is a solution of the static vacuum equations, which is non-flat; (x is essentially the point of maximal curvature concentration). Hence, for instance by Theorem 1.1, either (N, g_{b}) is a super-trivial solution or it is the (isometrically doubled) Schwarzschild metric, with non-vanishing potential function u. Thus, we are interested in considering only if the latter possibility occurs.

If the blow-up limit (N, g_{b}) is the Schwarzschild metric, then of course one must have M = M_{1}#M_{2}, with the Schwarzschild neck S^{2} × ℝ joining the two factors. This situation occurs in all versions of the construction in Example 2 in §6.2. However, for the construction in §6.1, it occurs only in the situation of Remark 6.1(ii), where a sum of two Green’s functions is used w.r.t. the conformal Laplacian on the standard (S^{3}, g_{can}).

We may assume that M_{2} is hyperbolic, and M_{1} is either hyperbolic or is a manifold satisfying σ(M_{1}) ≥ 0. In the former case, M_{o} is the one point union at x of M_{1} and M_{2}, while in the latter case, M_{o} = M_{2} and M_{1} is collapsed to X. (Note that one may have X = {x}; for instance, this is necessarily the case for the construction in Remark 6.1(ii), which requires N = M_{1} with σ(M_{1}) > 0).

It turns out that if M_{1} is hyperbolic, then the Schwarzschild neck (N, g_{b}) must be a super-trivial solution, for any sequence {g_{i}} satisfying the hypotheses above. On the other hand, if M_{1} is any graph manifold, (in particular σ(M_{1}) ≥ 0), and {g_{i}} collapses M_{1} with bounded curvature away from the neck to X, then the limit solution is non-trivial. The limit potential function u is given by the Schwarzschild potential, odd under reflection through the event horizon, as in (6.20). The construction in Theorem 3.10 produces this non-trivial blow-up limit.

For the remaining possible situations, where σ(M_{1}) is not a graph manifold, or {g_{i}} volume collapses M_{1} to a lower dimensional space, but the collapse is with unbounded curvature away from the neck, the triviality or non-triviality of the solutions is unknown. (Note this situation includes the construction in §6.1, Remark 6.1(ii)).

We do not give a complete proof of these statements here, since essentially no new ideas are needed, and since these statements will not be used. Instead, we just sketch the ideas involved.

When M_{1} is a graph manifold and {g_{i}} collapses M_{1} with bounded curvature, this statement can be deduced quite easily from the work in §6.5, in particular from Proposition 6.9. In this case, in fact ξ → 0, z^{T} → 0 and u → 1 in L^{2}(M, g_{i}); c.f. Remark 6.10(ii). Note that M_{1} becomes invisible, in terms of volume and total scalar curvature, in {(M, g_{i})}.

On the other hand, if M_{1} is assumed hyperbolic, then u cannot converge to 1 in L^{2}, essentially for the reasons discussed following (6.19). One may then prove that u_{i} is approximately 0 for i large on the Schwarzschild neck by using the minimizing property Proposition 2.7 for L^{∗}u, together with the existence of non-trivial solutions ξ, L^{∗}f to (6.25), (6.27) and (6.34)-(6.35) on M_{o} \ {x}, as discussed following Proposition 6.6.

§6.4. Example 3. In this next example, we let M = S^{2} × S^{1}, and consider a maximizing family {g_{i}} of Yamabe metrics on M with vol_{g_{i}} M = vol S^{3}(1). It is known that

σ(S^{2} × S^{1}) = σ(S^{3}).  (6.42)

In fact, see [Kb1,2], [Sc2], there are conformally flat, spherically (i.e. S^{2}) symmetric Yamabe metrics g_{i}, thus of the form

g_{i} = dt^{2} + f_{i}^{2}(t)ds_{S^{2}}^{2},  (6.43)

with s_{g_{i}} → σ(S^{3}). As noted in [La], and as in Example 2, setting h_{i} = f′_{i}, one computes

L^{∗}(h_{i}) = 0,  (6.44)

w.r.t. the metrics g_{i} globally on M. In fact, c.f. [La],

Ker L^{∗} =< h_{i} >,

on (M, g_{i}). Again from Proposition 3.0, the 4-manifolds X = M ×_{h_{i}} S^{1} are Einstein, with scalar curvature s_{X} → 12, the scalar curvature of S^{4}(1). (These metrics have cone singularities along two totally geodesic 2-spheres in X, corresponding to the locus where h_{i} = 0).

As i → ∞, the manifolds (S^{2}×S^{1}, g_{i}) converge to the space (Z, g_{o}), where Z = S^{3}/{p} ∼ {−p} is the 3-sphere with two antipodal points identified, and g_{o} is the canonical metric of constant curvature and volume 1, see [An1], [Sc2]. The convergence is smooth on the complement of the point {p} ∼ {−p}, and the functions h_{i}, for an appropriate normalization, converge to an eigenfunction h of the Laplacian on S^{3}, with eigenvalue 3, with h_{i}(p) → 1, h_{i}(−p) → −1.

Now consider the equation (2.10) on (M, g_{i}), i.e.

L^{∗}u + ξ = −\frac{s}{3}g.  (6.45)

The function u = u_{i} is only determined up to functions in Ker L^{∗}. Using suitable multiples of the functions h_{i} in (6.44), we may arrange that

||u_{i}||_{L^{2}} → ∞, as i → ∞.  (6.46)

(We note that it may well be necessary to use the functions h_{i} in this way; there may be representatives u′ ∈ {u + Ker L^{∗}} which are uniformly bounded in L^{2}). In case (6.46) holds, as discussed in §3.2-§3.4, the construction of the buffered blow-up in Theorem 3.10 requires that u = u_{i} be renormalized by its maximum. Thus, setting ū = \frac{u}{sup u}, we obtain from (6.45) that

L^{∗}ū_{i} → 0.  (6.47)

Then in fact ū_{i} approaches the function h_{i} in (6.44) and {ū_{i}} limits on the eigenfunction h of the Laplacian as above. If one performs the descent construction of Theorem 3.10 on {ū_{i}}, one obtains as blow-up limit the Schwarzschild metric (0.17) doubled across the totally geodesic boundary S^{2}. In the blow-up, the functions ū_{i} limit on the potential function u of the Schwarzschild metric, and the harmonic function u is odd w.r.t reflection in S^{2}. At one end, u → +1 while u → −1 at the other end, corresponding to the two points {p} and {−p} respectively in the limit (Z, g_{o}).

Thus in this simple example, the blow-up limit obtained from Theorem A(II) mirrors exactly the degeneration of the sequence (M, g_{i}).

§6.5. Example 4. The discussion in §6.1 and §6.2 raises the question of what are the simplest kinds of degenerations of Yamabe metrics which are modeled on non-trivial solutions of the static vacuum equations, (leaving Example 3 aside). Thus, let M_{1} and M_{2} be closed hyperbolic 3-manifolds and M = M_{1}#M_{2}. Based on a modification of Example 2, we construct a sequence {g_{i}} of Yamabe metrics on M which crush the essential 2-sphere in M, converge smoothly almost everywhere w.r.t. volume to the hyperbolic metrics on M_{1} and M_{2}, and for which blow-up limits obtained via Theorem 3.10 are non-trivial solutions to the static vacuum equations. It is interesting to compare the construction below with the work of O. Kobayashi in [Kb1, Thm.2].

First, return to the construction in Example 2 on each manifold M_{k}, k = 1, 2. This gives a metric g_{a} on M_{k} \ B_{x_{k}}(a), with totally geodesic, constant curvature boundary S = S^{2}(a), which is hyperbolic outside the ball B_{x_{k}}(ε) and is “Schwarzschild-like” in the neck A(a, ε/2). As in Examples 1,2 note that the curvature of g_{a} is uniformly bounded in A(ε/2, ε) provided ε ≥ a^{1/3}. As previously, we assume ε >> a^{1/3}, so that the sectional curvature of g_{a} is very close, (depending only on ε/a^{1/3}) to −1 on A(ε/2, ε), and the scalar curvature of g_{a} is almost −6 everywhere.

Now instead of identifying M_{k} \ B_{x_{k}}(a) along their isometric boundaries, we extend g_{a} past S^{2}(a) with the metric g on S^{2} × ℝ^{+} in (6.15). The resulting metric, still called g_{a}, is a complete metric on M̃_{k} ≡ M_{k} \ B_{x_{k}}(a) ∪ S^{2} × ℝ^{+}. Note that this metric is conformally equivalent, in a natural way, to a metric on M_{k} \ {x_{k}} which extends smoothly over {x_{k}}, so that (M̃_{k} ∪ {x_{k}}, g_{a}) is conformally compact. (Of course (M_{k} \ B_{x_{k}}(a), g_{a}) cannot be conformally compactified by adding a point). The blow-up limit of (M̃_{k}, g_{a}) at or near x_{k} as a → 0 is the complete, isometrically doubled Schwarzschild metric. The end of (M̃_{k}, g_{a}) is asymptotically hyperbolic. In fact, just as before on the other ‘outer’ side of S^{2}(a) in M̃_{k}, the ‘inner’ annulus A_{i} = A(ε/2, ε) ⊂ (S^{2} × ℝ^{+}, g_{a}) is almost isometric to an annulus A_{−1}(ε/2, ε) in H^{3}(−1) for ε >> a^{1/3}. In particular, the curvature of g_{a} is bounded in both the inner annulus A_{i} ⊂ S^{2} × ℝ^{+} and the (isometric) ‘outer’ annulus A_{o} ⊂ M_{k}.

To join the two inner annuli A_{i}, we choose a metric γ_{a} on G_{a} ≡ S^{3}\ (B_{1}∪B_{2}), B_{j} a 3-ball, so that γ_{a} is isometric to (A_{i}, g_{a}) near its boundary. Thus, the metrics (M̃_{1}, g_{a}) and (M̃_{2}, g_{a}) can be glued along (S^{3}\B_{1}∪B_{2}), γ_{a}) to give a smooth family of metrics, still denoted g_{a}, on M = M̃_{1}∪G_{a}∪M̃_{2} = M_{1}#M_{2}. The gluing metric γ_{a} is chosen to have curvature z_{γ_{a}} uniformly bounded as a → 0, scalar curvature s_{γ_{a}} → −6 everywhere as a → 0, and to be volume collapsing, so that

vol_{γ_{a}} G_{a} → 0, as a → 0.  (6.48)

Of course since the metrics g_{a} are not collapsing near the boundaries ∂B_{i} of G_{a}, the collapse of γ_{a} takes place on a scale much larger than ε. We sketch the construction of such metrics.

Construction of glueing metrics: View S^{3} as the union of two solid tori D^{2} × S^{1}, glued along the torus boundary T^{2} by interchanging the two circles in T^{2}. Consider a complete warped product metric on D^{2} × S^{1} of the form

h = g_{D^{2}} + f^{2}dθ^{2},  (6.49)

where f is a positive function on D^{2}. We assume that outside a compact set, h is isometric to a rank 2 hyperbolic cusp, so that in particular f = f(r) = e^{−r} for r large.

It is a standard fact that h may be chosen to satisfy

∫ s_{h} dV_{h} < 0,  (6.50)

c.f. [Bes, Thm.4.32] for example. Now let h̄ be a Yamabe metric conformal to h, so that h̄ = ψ^{4} · h. It is also quite standard, (c.f. [AM, Thm.C] for a proof), that (6.50) and the assumptions on the asymptotic form of h imply that h̄ exists and is complete, with ψ uniformly bounded away from 0 and ∞. Possibly after a rescaling, the scalar curvature s̄ of h̄ satisfies s̄ = −6.

Now we claim that the metric h̄ is also a warped product, i.e. is invariant under the S^{1} action on D^{2} × S^{1}. To see this, let F = F_{θ} be an isometry of h, (from the S^{1} family). We have

F^{∗}h̄ = (F^{∗}ψ)^{4} · (F^{∗}h) = (F^{∗}ψ)^{4} · h = v^{4} · h̄,

where v = (F^{∗}ψ)/ψ is a bounded function. The metrics F^{∗}h̄ and h̄ are thus conformal Yamabe metrics, of the same scalar curvature −6 and volume. Hence v satisfies the equation (1.12), i.e.

−6v^{5} = −8Δv − 6v,

w.r.t. the h̄ metric. The maximum principle then implies that v ≡ 1. Thus, F is also an isometry of h̄. It follows that ψ is a function on D^{2} and the defining equation (1.12) becomes, on D^{2},

8Δ_{D^{2}}ψ + 8 < ∇ψ , ∇logf >= 6ψ^{5} + sψ.  (6.51)

Since ψ is bounded and s = −6 outside a compact set, the maximum principle applied to (6.51) shows that ψ(x) → 1 as x → ∞ in D^{2} × S^{1}.

It follows that h̄ is also of the form (6.49), with f replaced by f̄, and is asymptotic to a rank 2 hyperbolic cusp at infinity.

Now observe that the full curvature of h̄ is unchanged when the length of the S^{1} factor is changed. In fact, for any α > 0 (small), the metrics

h̄_{α} = ḡ_{D^{2}} + (αf̄)^{2} dθ^{2},  (6.52)

are locally isometric to h̄. Note that the length of the S^{1} fiber is asymptotic to αe^{−r}.

Next, for R sufficiently large depending on the choice of α, on the annulus A(R, 2R), we may bend the metric ḡ_{D^{2}} ∼ dr^{2} + e^{−2r}dϕ^{2} slowly on A(R, 2R) so that on A(2R − 1, 2R), it has the form dr^{2} + (αf̄)^{2} dϕ^{2}. Here we assume that both θ and ϕ are parameters in [0, 2π]. The resulting metric h̃_{α} has uniformly bounded curvature, independent of α and R, scalar curvature arbitrarily close, depending only on R, to −6, and is almost isometric to a rank 2 hyperbolic cusp on A(2R − 1, 2R).

Finally, on the annulus A(2R, 4R), the function αf̄ is again bent slowly so that at the boundary S(4R), αf̄(4R) ∼ αe^{−4R}, f̄′(4R) = 0 and f̄ extends smoothly as an even function under reflection through 4R. Again this may be done so that the curvature is uniformly bounded and the scalar curvature is arbitrarily close to −6. Call the resulting metric on D^{2} × S^{1} again h̃_{α}.

Now the boundary S(4R) is a totally geodesic, flat, and square torus T^{2}, in the sense that the generators dϕ and dθ are orthogonal and of the same length. Hence the metric h̃_{α} may be doubled across the boundary T^{2}, with an interchange of the S^{1} factors, to give a smooth metric on S^{3}. This is the metric γ_{a}, where α in (6.52), essentially the maximal length of the S^{1} fiber in (6.52), is relabeled to a = a(α), the size of the core S^{2} in (M_{k}, g_{a}) above. We require a << α, in fact ε << a, but a → 0 implies α → 0.

By choosing R = R_{α} suitably, the resulting family of smooth metrics γ_{a} on S^{3} has scalar curvature converging smoothly to −6 as a → 0, has uniformly bounded curvature, and is volume collapsing in the sense of (6.48). Observe that this volume collapse requires

diam_{γ_{a}} S^{3} ∼ 2R → ∞, as a → 0.  (6.53)

Now the inner annuli (A_{i}(ε/2, ε), g_{a}) in each M̃_{k} are almost isometric to hyperbolic ε-annuli. For ε << α, the metrics γ_{a} on the geodesic ball B_{p}(ε) contained in each D^{2}×S^{1} ⊂ S^{3} are fixed, independent of a or α; here p = (p_{o}, θ), where p_{o} is the ‘center’ of D^{2}. (The metric on D^{2} is fixed, only the length of the fiber S^{1} changes with a). Hence, just as with the first or outer glueing as in Example 2, the metric γ_{a} may be smoothly matched to (A_{i}, g_{a}), keeping the curvature uniformly bounded and the scalar curvature arbitrarily close to −6, (c.f. the local deformation in Remark 6.1(iii)). This completes the construction of the glueing metrics.

Finally, as in Examples 1, 2, let ḡ_{a} be the Yamabe metric conformal to g_{a} with the same volume on M, so that ḡ_{a} = w^{4} · g_{a}. Since the scalar curvature of g_{a} approaches −6 everywhere as a → 0, as in (6.12), w → 1 pointwise as a → 0. Thus, the metric ḡ_{a} is C^{o} close to g_{a}. Further, from the defining equation (1.12), |Δw| → 0 pointwise as a → 0. Using L^{2} estimates for the elliptic equation (1.12) on any geodesic ball B about q, of radius ρ(q), one obtains w.r.t. the g_{a} metric,

∫_{B} |D^{2}w|^{2} << max (1, ∫_{B} |z|^{2}) as a → 0,  (6.54)

provided ρ(q) ≤ 1. Of course ḡ_{a} is smoothly close to g_{a} on M \ B_{x_{1}}(ε) ∪ B_{x_{2}}(ε)). This completes the construction of the Yamabe metrics {ḡ_{a}} on M.

The justification for this construction comes from the following result.

Proposition 6.9. On the family (M, ḡ_{a}),

||z^{T} ||_{L^{2}} → 0 as a → 0.  (6.55)

In particular, ξ → 0 in L^{2}, and δ → 0.

From Proposition 4.1 for instance, it follows that

u = u_{a} → 1 in L^{2},  (6.56)

and hence, (c.f. Proposition 4.2), one may carry out the descent construction in Theorem 3.10 to obtain non-trivial blow-up limit solutions of the static vacuum equations. In fact for an initial sequence of points {x_{i}} in either M_{k} \ B_{x_{k}}(a), with x_{i} → x_{k}, u_{i}(x_{i}) → 1, and a = a(i) → 0, the maximal blow-up limit (N, g_{o}), c.f. §5.1, based at a buffered sequence {y_{i}} obtained by u-descent from {x_{i}} is the complete doubled Schwarzschild metric. The limit potential function is u = ±(1 − \frac{2m}{t})^{1/2}, c.f. (0.17).

Proof of Proposition 6.9. To prove (6.55), we use the characterization (2.41) of ||z^{T} ||_{L^{2}}, i.e.

∫_{M} |z^{T} |^{2} = inf ; ∫ ϕ=0 ∫_{M} |z − L^{∗}ϕ|^{2}.  (6.57)

Of course (6.57) is taken w.r.t. the ḡ_{a} metric. However, using the fact that ḡ_{a} is C^{o} close to g_{a}, and smoothly close away from M \ (B_{x_{1}}(ε) ∪ B_{x_{2}}(ε)), together with (6.54), it suffices to estimate the right hand side of (6.57) w.r.t. the g_{a} metric. To see this, refer to the expansion (2.42). Using standard formulas, c.f. [Bes, Ch.1J], under the conformal change from g_{a} to ḡ_{a}, all terms in (2.42) differ by arbitrarily small quantities as a → 0, except the term containing |z|^{2}, which differs by a term on the order of |D^{2}w|^{2}, where w is the conformal factor. By (6.54), this is small in B_{x_{1}}(ε) ∪ B_{x_{2}}(ε) compared with the dominant |z|^{2} term in (2.42).

Of course, since the metric g_{a} is quite explicit, it is easier to estimate (6.57) w.r.t. g_{a} than w.r.t. ḡ_{a}. All the estimates to follow are then on (M, g_{a}).

The function ϕ = ϕ_{a} is defined on the various pieces of (M, g_{a}) as follows. First, on each M_{k}\B_{x_{k}}(ε), since z = 0, we set ϕ ≡ 0, so that

∫_{M_{k}\B_{x_{k}} (ε)} |z − L^{∗}ϕ|^{2} = 0.  (6.58)

Next, on the (doubled) Schwarzschild-like neck N_{a} joining the inner and outer annuli A(ε/2, ε) on each M̃_{k}, let

ϕ = h − 1,  (6.59)

where h = h_{a} is the function from (6.16)-(6.17). Note that h is odd under reflection through the core S = S^{2}(a) in each M_{k}. Since L^{∗}h = 0 w.r.t. the metric g_{a}, L^{∗}ϕ = r on N_{a}. Hence, since vol N_{a} → 0 as a → 0, and the scalar curvature is uniformly bounded,

∫_{N_{a}} |z − L^{∗}ϕ|^{2} → 0, as a → 0.  (6.60)

To examine the behavior on the outer or first glueing annulus A_{o} = A(ε/2, ε) ⊂ M_{k}, the metric g_{a} has uniformly bounded curvature on A_{o} as a → 0. Now from the discussion in Example 2, in A_{o},

h ∼ cosh t = 1 + \frac{1}{2}t^{2} + o(t^{2}),  (6.61)

so that ϕ ∼ 0. Further, since h ∼ h(t),

|D^{2}h| ∼ |h′D^{2}t + h′′(dt)^{2}| ≤ C,  (6.62)

uniformly in A_{o} as a → 0, by (6.61). Hence the function ϕ may be smoothed in A_{o} so that ϕ ≡ 0 near the outer boundary S(ε) of A_{o}, keeping D^{2}ϕ uniformly bounded. Since vol A_{o} → 0 as a → 0, it follows that (6.60) holds also over A_{o}.

Finally, in the glueing region G_{a} joining M̃_{1} and M̃_{2}, and containing the inner annuli A_{i}, the curvature of g_{a} is uniformly bounded while vol G_{a} → 0, so that

∫_{G_{a}} |z − L^{∗}ϕ|^{2} ≤ δ_{a} + 2 ∫_{G_{a}} |L^{∗}ϕ|^{2} ≤ δ_{a} + c ∫_{G_{a}}(|D^{2}ϕ|^{2} + ϕ^{2}),  (6.63)

where δ_{a} → 0 as a → 0. Now in the inner annulus A_{i} = A(ε/2, ε) ⊂ S^{2} × ℝ^{+}, by considerations similar to (6.61), ϕ ∼ −2, while as in (6.62), |D^{2}ϕ| is uniformly bounded. Hence, using (6.53), we may extend ϕ from a neighborhood of both components of ∂G_{a} into G_{a} - for instance as a function of a suitable smoothing of the distance function to x_{1} (or x_{2}) - so that ϕ and D^{2}ϕ remain uniformly bounded and so that, (most importantly),

∫_{M} ϕdV_{g_{a}} ∼ ∫_{M} ϕdVḡ_{a} = 0.  (6.64)

Again, since vol G_{a} → 0, it follows that the right side of (6.63) goes to 0 as a → 0. These estimates together then prove (6.55).

■

The limit of (M, ḡ_{a}) as a → 0 is the union of the two hyperbolic manifolds M_{1} and M_{2}, glued along a lower dimensional length space of infinite diameter and zero volume. In particular, M_{1} and M_{2} are infinitely far apart in the limit. The pointed Hausdorff limits of (M, ḡ_{a}, q_{a}) as a → 0 are complete subsets of this structure, based at some limit point q = lim q_{a}.

Of course, the limit here is quite different from the limits of the sequences in §6.1-§6.3.

Remark 6.10. (i) Observe that in obtaining (6.55), rather strong use has been made of the fact that the limit metrics on M_{1} and M_{2} are hyperbolic. Although the construction in Example 2 is valid quite generally, (c.f. Remark 6.2(i)), and does not require the limits to be hyperbolic, (6.55) will no longer hold for constructions as above with non-hyperbolic limits.

(ii). In addition to taking connected sums of hyperbolic manifolds, one may carry out this construction on manifolds of the form N#M, where N is any closed oriented graph manifold, (and hence σ(N) ≥ 0), compare with Remark 6.2(ii).

Namely, carry out the construction above on M_{2} = M and replace the glueing region G_{a} = S^{3}\(B_{1}∪ B_{2}) by N \ B. Just as S^{3} admits metrics γ_{a} satisfying (6.48) and the curvature conditions preceding (6.48), so does the manifold N. The remainder of the argument then proceeds as before. Proposition 6.9 holds as before also.

As in Remarks 6.1 and 6.2, since the construction is essentially local, it can be iterated (arbitrarily) many times.

Remark 6.11. It is worthwhile to mention explicitly that all known constructions or examples of degenerating sequences of Yamabe metrics on a given 3-manifold M have blow-up limits which do have one common feature; namely, they are all scalar-flat and asymptotically flat.

7. Palais-Smale Sequences for Scalar Curvature Functionals.

The discussion through §5 applies to quite general sequences of Yamabe metrics, and has made no assumptions that the sequence {g_{i}} is a maximizing sequence for S|_{C} or even a sequence for which S(g_{i}) approaches a critical value of S|_{C}. Of course, we are most interested in understanding the degeneration of a maximizing sequence of Yamabe metrics, and one would expect that such a sequence may have some restrictions on its behavior not valid for general sequences.

In this section, we examine more closely the structure of the space of metrics 𝕄 and its completeness properties w.r.t. natural metrics. This is then used to understand the existence of Palais-Smale sequences for natural functionals on 𝕄 or its subvarieties. In particular, the differences in the behaviors of the examples in §6 can be understood from this viewpoint.

§7.1. The space 𝕄 is, of course, not complete with respect to the L^{2} or L^{2,2} metrics. We study its completion w.r.t. the L^{2,2} norm, (and later the T^{2,2} norm). The L^{2,2} norm (1.4) on the collection {T_{g}𝕄} of tangent spaces generates a length metric L^{2,2} on 𝕄, by defining L^{2,2}(g_{1}, g_{2}) to be the infimum of the lengths of curves joining g_{1} to g_{2}.

Let 𝕄_{L^{2,2}} be the Cauchy-completion of 𝕄 w.r.t. the L^{2,2} metric, so an (ideal) point in 𝕄_{L^{2,2}} is a limit point of a Cauchy sequence w.r.t. the L^{2,2} metric. Let

𝕄̄ = 𝕄_{L^{2,2}} ∩ C^{0}(𝕄),  (7.1)

where C^{0}(𝕄) is the space of continuous Riemannian metrics on M.

By work in [E], the space 𝕄̄ may be identified with the space of Riemannian metrics on M which, with respect to a fixed smooth coordinate atlas, have local expressions which are L^{2,2} functions of the local coordinates. The tangent space T_{g}𝕄̄ is naturally identified with the Hilbert space of symmetric bilinear forms on M, locally in L^{2,2}, with inner product given as in (1.4).

Let 𝕄_{o} ⊂ 𝕄 be the space of metrics having a fixed volume form dV, of total volume 1, and 𝕄_{1} ⊂ 𝕄 the space of metrics of total volume 1. Similarly, let 𝕄̄_{o}, and 𝕄̄_{1} be the intersection of the L^{2,2}- completion of 𝕄_{o} or 𝕄_{1} with C^{0}(𝕄_{o}) or C^{0}(𝕄_{1}) respectively. The spaces 𝕄̄, 𝕄̄_{o} and 𝕄̄_{1} are infinite dimensional Hilbert manifolds in the topology generated by the L^{2,2} norm, c.f. [E].

It is important to note that the spaces 𝕄̄, 𝕄̄_{o}, 𝕄̄_{1} need not apriori be (Cauchy) complete in the L^{2,2} metric. By definition, of course 𝕄_{L^{2,2}} is complete. However, the symmetric bilinear forms in such a completion may not be positive definite, and thus not metrics. Thus, the restriction that 𝕄̄ be contained in C^{0}(𝕄) implies that 𝕄̄ might not apriori be complete.

To understand the completeness of these spaces, following the discussion in [E], we first decompose the space of smooth metrics 𝕄 as

𝕄 = Vol(M) × 𝕄_{o}, g = ϕ · h  (7.2)

where Vol(M) is the space of volume forms on M, h is an arbitrary metric in 𝕄_{o} and ϕ is the ratio of the volume forms of g and h to the power 2/n = 2/3. Note that the metric g in (7.2) is conformal to h, so that 𝕄_{o} gives a representation of the space of conformal structures on M. This should be compared with the splitting of 𝕄 into the space of conformal classes and the space C of Yamabe metrics. Here we recall a well-known result of Moser [Mo] that any metric in 𝕄_{1} may be pulled back by a diffeomorphism of M to a metric in 𝕄_{o}. In particular, for any unit volume Yamabe metric g ∈ C_{1}, there is a diffeomorphism ψ of M such that ψ^{∗}g ∈ 𝕄_{o}.

Given a fixed background metric g_{o} ∈ 𝕄_{o}, any metric h ∈ 𝕄_{o} may be written as

h = g_{o} · e^{X},  (7.3)

where X is trace-free w.r.t. g_{o}; here we are using the metric g_{o} to identify bilinear forms with linear maps.

Lemma 7.1. The space 𝕄̄_{o} is (Cauchy) complete.

Proof. Let {g_{i}} be a Cauchy sequence in 𝕄̄_{o} w.r.t. the L^{2,2} metric. Then by definition, c.f. (7.1), the sequence {g_{i}} converges to an element g ∈ 𝕄_{L^{2,2}}, i.e. a symmetric bilinear form on M with local component functions in the L^{2,2}-completion of C^{∞}(M). By Sobolev embedding g_{i} → g in the C^{0} topology. Since the volume forms dV_{g_{i}} are fixed, it follows that dV_{g} = dV_{g_{i}}. In particular, dV_{g} is a (continuous) positive 3-form on M. However, the symmetric bilinear forms in 𝕄_{L^{2,2}} are necessarily positive semi-definite. It follows that g is a positive definite form and hence in 𝕄̄_{o}.

■

It follows from the Hopf-Rinow theorem in infinite dimensions that 𝕄̄_{o} is also geodesically complete w.r.t. the L^{2,2} metric, i.e. L^{2,2} geodesics in 𝕄̄_{o} exist for all time. We note that 𝕄_{o} itself is geodesically complete, (but of course not Cauchy-complete), w.r.t the L^{2} metric, c.f. [E, Thm.8.9]; in fact the L^{2} geodesics in 𝕄_{o} are given by the simple expression, c.f. (7.3),

g(t) = g · e^{tA} ⊂ 𝕄_{o},  (7.4)

where A ∈ T_{g}𝕄_{o}, so that A is a smooth symmetric bilinear form, trace-free w.r.t. g. In particular, these geodesics exist for all time. Further, the L^{2} exponential map is a diffeomorphism onto 𝕄_{o}.

On the other hand, the full space 𝕄̄ is not complete with respect to the L^{2,2} metric, nor geodesically complete w.r.t. the L^{2} metric. With regard to the latter, in contrast to (7.4), the geodesics of 𝕄 with respect to the L^{2} metric do not exist for all time, but may leave the space 𝕄 in finite time. In [FG], the L^{2} geodesic of 𝕄, with initial position (μ, g) and initial velocity (w, A), c.f. (7.3) and (7.4), is calculated to be given by

g(t) = (v(t)^{2} + n^{2}t^{2})^{2/3}g · exp (\frac{tan^{−1}(nt/v)}{n}A) ,  (7.5)

where v(t) = 1 + \frac{1}{2}(\frac{w}{μ})t and n = \frac{1}{4}(3 tr A^{2})^{1/2}.

A brief inspection, as noted in [FG], shows that these curves escape from 𝕄 in finite time, (i.e. the form g(t) is no longer positive definite), if the initial velocity matrix A vanishes somewhere on M. For example, in the volume or conformal directions, when A ≡ 0, it is apparent from (7.5) that g(t) becomes degenerate, i.e. non-positive definite, in finite time.

Although we will not carry it out here, it is not difficult to verify that there are curves g(t), t ∈ [0, 1), of the form (7.5), with A ≡ 0, of finite length in the L^{2,2}, which degenerate as t → 1. In particular, the L^{2} norm of the curvature blows up as t → 1.

On the other hand, if the matrix A never vanishes on M, i.e., if A(p) ≠ 0, for all p ∈ M, then the geodesic (7.5) with initial velocity (w, A) continues in 𝕄 for all time, for any initial metric (μ, g). Thus, in most all directions in 𝕄̄, the L^{2} geodesic (7.5) exists for all time.

Given this background, we now discuss the main result of this section. Let F : 𝕄̄_{o} → ℝ^{+} be a C^{1} functional on 𝕄̄_{o}, which is thus bounded below, We only consider ’natural’ functionals, in the sense that F(ϕ^{∗}g) = F(g), for any ϕ ∈ Diff(M). For example, it is easy to see that the total scalar curvature functional S, or the L^{p} norm of the scalar curvature S^{p}, for 1 < p ≤ 2, extends to a C^{1} functional on 𝕄̄_{o} or 𝕄̄_{1}.

Theorem 7.2. Suppose F : 𝕄̄_{o} → ℝ_{+} is a C^{1} natural functional, as above, and let {γ_{i}} ∈ 𝕄_{o} be a minimizing sequence for F. Given any ε > 0, there is another minimizing sequence {g_{i}} ∈ 𝕄_{o} for F, with L^{2,2}(g_{i}, γ_{i}) ≤ ε, such that ∇F(g_{i}) → 0, as i → ∞ in the dual (L^{2,2})^{∗} norm, i.e.

|∇F|^{∗}(g_{i}) ≡ sup_{|α|_{L^{2,2}=1}} |∫_{M}⟨∇_{g_{i}}F, α⟩_{g_{i}}dV_{g_{i}} | → 0, as i → ∞,  (7.6)

for α = α_{i} ∈ T_{g_{i}} 𝕄̄_{o}.

Proof. For otherwise, given any ε > 0, there exists c = c(ε) > 0 such that, (for some subsequence),

|∇F|^{∗}(g_{i}) ≥ c, as i → ∞.  (7.7)

for all g_{i} ∈ 𝕄̄_{o} such that

L^{2,2}(g_{i}, γ_{i}) ≤ ε.  (7.8)

Let γ_{i}(t) be a smooth curve in 𝕄̄_{o}, with γ_{i}(0) = γ_{i}, |\frac{d}{dt}γ_{i}(t)| = 1, (i.e. γ_{i}(t) is parameterized by arclength in the L^{2,2} metric), and

∫_{M}⟨∇_{γ_{i}(t)}F, \frac{d}{dt}γ_{i}(t)⟩dV_{γ_{i}(t)} ≤ −\frac{c}{2}.  (7.9)

Since 𝕄̄_{o} is an infinite dimensional manifold and ∇F is continuous, it is obvious from (7.7) that such curves exist for t sufficiently small, say for 0 < t ≤ t_{o} = t_{o}(i); one could take for instance γ_{i}(t) to be the L^{2} geodesic (7.4) in the direction −∇_{γ_{i}}F, but reparametrized w.r.t. L^{2,2} arclength.

If t_{o} < ε, we just repeat the process above starting at γ_{i}(t_{o}). Since the L^{2,2} metric is complete on 𝕄̄_{o}, it is clear that one obtains in this way a continuous piecewise smooth curve γ_{i}(t) in 𝕄̄_{o}, satisfying (7.9) for 0 ≤ t ≤ ε. By definition, (7.9) then gives

\frac{d}{dt}F(γ_{i}(t)) = ∫_{M}⟨∇_{γ_{i}(t)}F, \frac{d}{dt}γ_{i}(t)⟩dV_{γ_{i}(t)} ≤ −\frac{c}{2}.  (7.10)

for all t ∈ [0, ε]. By integration of (7.10), it follows that

F(γ_{i}(t)) ≤ F(γ_{i}) − \frac{c}{2} · t.  (7.11)

The fact that (7.11) is valid for all t ≤ ε, contradicts the fact that {γ_{i}} is a minimizing sequence for F. Hence (7.7) cannot hold in conjunction with (7.8) and the result follows.

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A sequence of metrics {g_{i}} satisfying (7.6) will be called a Palais-Smale sequence for F, w.r.t. the L^{2,2} norm. Note that this definition corresponds to one of the hypotheses in the well-known Condition C of Palais-Smale [PS]. Of course, the condition (7.6) should not be confused with Condition C—a compactness condition which is much too strong to be of any use in this context.

Thus Theorem 7.2 implies that within any ε-neighborhood, w.r.t. the L^{2,2} metric, of a minimizing sequence for F, there exists a Palais-Smale minimizing sequence.

Since, by Moser’s theorem mentioned above, 𝕄_{1} is obtained from 𝕄_{o} by the action of the diffeomorphism group, and F together with (7.6) is diffeomorphism invariant, it follows that Theorem 7.2 holds for F considered as a functional F : 𝕄̄_{1} → ℝ^{+}, with α ∈ T 𝕄_{1}.

Remark 7.3. It is easy to see that Theorem 7.2 also holds for functionals on other natural submanifolds of 𝕄_{1}, obtained by diffeomorphims from 𝕄_{o}. For instance, at least in case σ(M) ≤ 0, so that C_{1} is an infinite dimensional submanifold of 𝕄_{1}, one may verify without difficulty that Theorem 7.2 holds for S|_{C_{1}}, (or −S|_{C_{1}}).

§7.2. Since the L^{2,2} metric is rather strong, the corresponding dual (L^{2,2})^{∗} = L^{−2,2} metric is rather weak. It is thus of interest to understand if the results above can be strengthened by use of a weaker norm than the L^{2,2} norm. Of course, the main issue in §7.1 is the Cauchy completeness of 𝕄̄_{o}.

In this respect, the following might be useful.

Lemma 7.4. The completion of 𝕄 w.r.t. the T^{2,2} norm (1.9), as in (7.1), is the same as its completion 𝕄̄ w.r.t. the L^{2,2} norm.

Proof. Let g_{i} be a T^{2,2} Cauchy sequence in 𝕄, converging to an element g in 𝕄_{T^{2,2}} ∩ C^{0}(M). Thus g is a continuous metric on M, and the local components g_{kl} of g in a smooth atlas on M are T_{g}^{2,2} functions of the local coordinates. It follows that Δ_{g}(g_{kl}) is locally in L^{2} on M. Hence, using the continuity of g, elliptic regularity theory, c.f. [GT, Thms.8.8,9.15], implies that g_{kl} is locally in L_{g}^{2,2}. Thus g ∈ 𝕄̄, which gives the result. ■

It is now easy to verify that all of the results above, in particular Theorem 7.2, hold also w.r.t. the weaker T^{2,2} norm; hence there exist (many) Palais-Smale sequences for F as above w.r.t. the T^{2,2} norm.

Let us apply this to the functional S|_{C_{1}}. If g_{i} is a Palais-Smale sequence of Yamabe metrics in C_{1}, then

∥∇_{g_{i}}(S|_{C_{1}})∥_{T^{−2,2}(T C_{1})} = sup_{|α|_{T^{2,2}}=1} |∫_{M}⟨z^{T} , α⟩dV_{g_{i}} | → 0,  (7.12)

where z^{T} = z_{g_{i}}^{T} and α ∈ T_{g_{i}}C_{1}. Now (unfortunately), the condition that α = α_{i} ∈ T_{g_{i}}C_{1} is a global condition on α, (as a tensor on (M, g_{i})), and hence it is difficult to obtain local information on z^{T} from (7.12). If (7.12) could be strengthened to hold for all α ∈ T_{g_{i}}𝕄_{1}, i.e. if {g_{i}} is strongly Palais-Smale in the sense of (4.6), then the estimate (7.12) does provide non-trivial local control. In particular, Propositions 4.1 and 4.2 hold in this situation. We do not know however if there exist such strongly Palais-Smale sequences on general 3-manifolds M.

With regard to the examples of §6, it is easily verified that all sequences in Example 4 are Palais-Smale for S|_{C_{1}} w.r.t. the T^{2,2} norm. In fact, such sequences are strongly Palais-Smale, in the sense of (4.6), since the full L^{2} norm of z^{T} goes to 0 by Proposition 6.9. Note here that of course ||z^{T} ||_{T^{−2,2}} ≤ ||z^{T} ||_{L^{2}} on {g_{i}}. Similarly, the sequences in Example 3 are strongly Palais-Smale, since the same, but in this case much simpler, reasoning proving Proposition 6.9 holds here also. On the other hand, Examples 1 and 2 are clearly not strongly Palais-Smale for S|_{C_{1}} w.r.t. T^{2,2}, since for instance lim_{i→∞}(z^{T})_{g_{i}} is non-zero on the limit (M_{o}, g_{o}), c.f. §6.3. It seems non-trivial to prove, although we certainly conjecture, that the sequences in Example 1 and 2 are not Palais-Smale for S|_{C_{1}}.

8. Appendix

In this Appendix, we prove Theorem 3.2. We break the proof into several parts. The first result states that the event horizon Σ must be non-empty.

Theorem A.1. The only complete solution (N, g, u) to the static vacuum Einstein equations with u > 0 everywhere is the flat solution on ℝ^{3} or a quotient ℝ^{3}/Γ, with u = const.

Proof. Since u is positive on the complete manifold N, the associated 4-manifold X^{4} = N^{3} ×_{u} S^{1} is Ricci-flat and complete, c.f. (1.14). Define the function h by

h = log u.

A straightforward calculation gives

Δ_{X}h = 0.  (A.1)

Suppose first that either u is bounded above, or bounded below away from 0. Thus, h has either a uniform upper or a uniform lower bound. The Liouville theorem of Yau [Yu] then implies that h = const., and thus u = const. > 0. The equations then immediately imply that the metric g is flat.

In general, since X^{4} is Ricci flat, the infinitesimal Harnack inequality of Cheng-Yau [CY, Theorem 6] gives the bound

sup_{B^{4}(r)} |∇(log v)| ≤ C · \frac{1}{r},

for any positive harmonic function v defined on a geodesic ball B^{4}(r) ⊂ X. In particular, by integration, this leads to the usual Harnack inequality

sup_{B^{4}(r)} v ≤ C · inf_{B^{4}(r)} v.  (A.2)

Now suppose u is not constant. Then applying the estimate (A.2) to the functions h − a and b − h, where a and b, depending on r, are chosen to make these functions positive on B^{4}(r) implies, by a well known technique due to Moser, c.f. [GT, Theorem 8.22], that u has oscillation growing at a definite power of r, as r → ∞. In fact, an observation of Cheng [Cg] is that if h is non-constant, then h must have at least linear growth, i.e. there is a constant C < ∞ such that, for r ≥ 1,

osc_{B^{4}(r)} h = sup_{B^{4}(r)} h − inf_{B^{4}(r)} h ≥ C · r,  (A.3)

where as above B^{4}(r) is the geodesic r-ball about some fixed point x_{o} ∈ X. Suppose first that

inf_{B^{4}(r)} h < −c_{1}r,  (A.4)

for some c_{1} > 0. It follows again from the Harnack inequality that for any r, there are points x_{r} ∈ S_{π(x_{o})}^{3}(r) ⊂ N such that u(x) ≤ c_{2}e^{−c_{1}·r}, for all x ∈ B_{x_{r}}^{3}(1), r large, for some constant c_{2} > 0; here π : X^{4} → N^{3} is the projection on the first factor and S^{3}(B^{3}) denote geodesic spheres (resp. balls) in N. We use this to estimate the volume of regions in X^{4}. Setting U_{x_{r}}(1) = π^{−1}(B_{x_{r}}^{3}(1)), we have

vol_{X^{4}} U_{x_{r}}(1) = ∫_{B_{xr} (1)} u dV ≤ c_{3} · vol B_{x_{r}}^{3}(1)e^{−c_{1}·r}.  (A.5)

To estimate vol B_{x_{r}}^{3}(1), it follows from the Harnack inequality as above that there is a constant C_{o}, independent of r, s.t.

C_{o}^{−1} ≤ \frac{sup_{B} u(x)}{inf_{B} u(x)} ≤ C_{o}.  (A.6)

where we set B = B_{x_{r}}^{3}(1) ⊂ U_{x_{r}}(1); here we are using the fact that u is invariant in the fiber or S^{1} factor. Consider the conformally equivalent metric g̃ = u^{2}g from (1.19). Thus, if u_{o} = inf_{B} u(x), then vol_{g} B ≤ u_{o}^{−3} volg̃ B. On the other hand, since distances in g̃ are at most C_{o}u_{o} times distances in g, it follows that B ⊂ Bg̃(C_{o} · u_{o}). Since from (1.20), Ricg̃ ≥ 0, the volume comparison theorem for Ricci curvature implies volg̃ B(C_{o} · u_{o}) ≤ \frac{4}{3}πC_{o}^{3}u_{o}^{3}. It follows there is a constant D, independent of r, s.t.

vol B_{x_{r}}^{3}(1) ≤ D,  (A.7)

so that from (A.5), we obtain

vol_{X^{4}} U_{x_{r}}(1) ≤ c_{4}e^{−c_{1}·r},  (A.8)

for some constant c_{4}. However, (A.8) contradicts the (relative) volume comparison principle. Namely, for Ricci flat 4-manifolds, and r ≥ 1, one has the bound

\frac{vol B_{x_{r}}^{4}(r)}{r^{4}} ≤ vol B_{x_{r}}^{4}(1) = vol U_{x_{r}}(1),

where the equality follows from the fact that the fiber S^{1} has exponentially small length near x_{r} ∈ N ⊂ X. Together with (A.8), this implies

vol B_{x_{r}}^{4}(r) ≤ c_{4}r^{4}e^{−c_{1}·r} → 0 as r → ∞,

which is of course impossible.

On the other hand, if (A.4) does not hold, then by (A.3), one has the estimate

sup_{B^{4}(r)} h > c_{5} · r,

for some c_{5} > 0. Hence, there are points x_{r} ∈ N ⊂ X near which the fiber S^{1} has exponentially large growth. Arguing in exactly the same way as above, from the volume comparison theorem for non-negative Ricci curvature, vol B_{x_{r}}(1) ≥ c_{6}r^{−3} and hence,

vol B_{x_{r}}^{4}(r) ≥ c_{7}r^{−3}e^{c_{5}·r},

which is again impossible by the relative volume comparison theorem on X.

It follows that h must be constant and thus (N, g) is flat.

■

We note the following compactness principle for solutions of the static equations.

Lemma A.2. Let (g_{i}, u_{i}) be a sequence of solutions to the static vacuum equations (0.16), defined on a geodesic ball B_{i} = B_{x_{i}}(1), (w.r.t. the metric g_{i}). Suppose

vol B_{i} ≥ v_{o} > 0, |r_{i}| ≤ Λ < ∞,  (A.9)

and

u_{i}(x_{i}) ≥ u_{o} > 0, u_{i}(y_{i}) > 0, ∀y_{i} ∈ B_{x_{i}}(1),  (A.10)

for some constants v_{o}, Λ, h_{o}. Then, for any d > 0, a subsequence of the Riemannian manifolds (B_{x_{i}}(1−d), g_{i}) converges, in the C^{∞} topology, modulo diffeomorphisms, to a limit manifold (B_{x}(1−d), g), with limit function u. The triple (B_{x}(1−d), g, u) is a smooth solution of the static vacuum equations.

Proof. The Cheeger-Gromov compactness theorem, c.f. Theorem 1.3 and references there, implies that a subsequence of (B_{i}, g_{i}) converges, in the C^{1,α} topology, to a limit manifold (B, g). The bounds (A.9) and (A.10) imply further that the positive harmonic functions u_{i} satisfy a uniform Harnack inequality in B_{i}(1 − d). If there is a uniform bound u_{i}(x_{i}) ≤ u^{o} < ∞, it follows that a subsequence of {u_{i}} converges in the C^{o} topology to a limit positive function u. If not, so u_{i}(x_{i}) → ∞, we renormalize u_{i} by setting ū_{i} = u_{i}/u_{i}(x_{i}). Then again the fact that ū_{i} is harmonic and the Harnack inequality imply that {ū_{i}} has a subsequence converging to a positive limit harmonic function u in B(1 − d). As noted in §1.3, since the metrics g_{i} satisfy an elliptic equation (0.16), the regularity theory for elliptic equations implies that one has C^{k,α} bounds for solutions, in terms of C^{1,α} bounds, i.e. all covariant derivatives of the curvature are bounded in terms of the bounds (A.9)-(A.10). Equivalently, one can pass to the Ricci flat 4-manifold N and use the Einstein equation to obtain C^{k,α} regularity, for any k. These estimates imply convergence in the C^{k} topology, and thus C^{∞} convergence.

Using elliptic regularity on the equations (0.16), the L^{∞} bound on |r| in (A.9) may be replaced by weaker bounds; for instance, it suffices to assume an L^{2} bound on r.

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The following Corollary gives an apriori estimate for the curvature of a solution away from the event horizon Σ. The proof is a standard consequence of Theorem A.1 and Lemma A.2.

Corollary A.3. Let (N, g, u) be a solution to static vacuum equations (0.16), and U ⊂ N a domain with smooth boundary on which u > 0. Then there is an (absolute) constant K < ∞, independent of (N, g) and U, such that for all x ∈ U,

|z|(x) ≤ \frac{K}{t(x)^{2}} ,  (A.11)

where t(x) = dist(x, ∂U).

Proof. The proof is by contradiction. Thus, assume that (A.11) does not hold. Then there are static vacuum solutions (N_{i}, g_{i}, u_{i}), smooth domains U_{i} ⊂ N_{i} on which u_{i} > 0 and points x_{i} ∈ U_{i} such that

t^{2}(x_{i}) · |z_{i}|(x_{i}) → ∞, as i → ∞.  (A.12)

Let t_{i} = t(x_{i}). Since it may not be possible to choose the points x_{i} so that they maximize |z_{i}| (over large domains), we shift the base points x_{i} as follows: choose s_{i} ∈ [0, t_{i}) such that

s_{i}^{2} sup_{B_{x_{i}}(t_{i}−s_{i})} |z_{i}| = sup_{s∈[0,t_{i})}(s^{2} · sup_{B_{x_{i}}(t_{i}−s)} |z_{i}|) → ∞, as i → ∞,  (A.13)

where the last estimate follows from (A.12), (set s_{i} = t_{i}). Let y_{i} ∈ B_{x_{i}}(t_{i} − s_{i}) be points such that

|z_{i}|(y_{i}) = sup_{B_{x_{i}}(t_{i}−s_{i})} |z_{i}|.  (A.14)

Further, setting s = s_{i}(1 − \frac{1}{k}), k > 1, in (A.13), one obtains the estimate

s_{i}^{2}|z_{i}|(y_{i}) ≥ s_{i}^{2} (1 − \frac{1}{k})^{2} · sup_{B_{x_{i}}(t_{i}−s_{i}(1− \frac{1}{k}))} |z_{i}| ≥ s_{i}^{2} (1 − \frac{1}{k})^{2} · sup_{B_{y_{i}}(s_{i}/k)} |z_{i}|,  (A.15)

so that

sup_{B_{y_{i}}(s_{i}/k)} |z_{i}| ≤ (1 − \frac{1}{k})^{−2} |z_{i}|(y_{i}),  (A.16)

Now rescale or blow-up the metric so that |z̃_{i}|(y_{i}) = 1 by setting g̃_{i} = |z_{i}|(y_{i}) · g, and consider the pointed sequence (U_{i}, g̃_{i}, y_{i}). We have

|z̃_{i}|(y_{i}) = 1,  (A.17)

and by (A.13) and scale invariance,

distg̃_{i}(y_{i}, ∂U_{i}) → ∞, as i → ∞.  (A.18)

Also, it follows from (A.16) that

|z̃_{i}|(x) ≤ C(distg̃_{i}(x, y_{i})).  (A.19)

We also normalize u by setting

ũ_{i}(x) = \frac{u(x)}{u(y_{i})},  (A.20)

and note by construction that ũ_{i} > 0 on U_{i}.

Suppose first that there exists v > 0 such that

volg̃_{i} B_{y_{i}}(1) ≥ v.  (A.21)

Then the volume comparison principle for bounded curvature and (A.21) imply that volg̃_{i} B_{x_{i}}(1) ≥ v(x_{i}), where v(x_{i}) depends only on distg̃_{i}(x_{i}, y_{i}). By the compactness of solutions, Lemma A.2, it follows that a subsequence converges, in the C^{∞} topology, to a limit solution (U_{∞}, g_{∞}, u_{∞}), which is complete and satisfies u_{∞} > 0 everywhere. (The minimum principle for harmonic functions implies that u_{∞} cannot vanish anywhere). By Theorem A.1, g_{∞} must be flat and u_{∞} constant. However, the smooth convergence guarantees that the equality (A.17) passes to the limit, contradicting the fact that g_{∞} is flat.

If (A.21) is not satisfied, so that volg̃_{i} B_{y_{i}}(1) → 0, as i → ∞, it follows that the sequence (U_{i}, g̃_{i}, y_{i}) is collapsing in the sense of Cheeger-Gromov on balls (B_{y_{i}}(R_{i}), g̃_{i}), where R_{i} → ∞ as i → ∞. In dimension 3, the structure theory of collapse implies that the collapse is along an injective F-structure. More precisely, the balls (B_{y_{i}}(R_{i}), g̃_{i}) have the structure of a Seifert fibration, with fibers that are injective in the fundamental group, c.f. [An2, §2,3] or [An II, §2]. Thus, one may pass to the universal cover of B_{y_{i}}(R_{i}). Since the universal covers no longer collapse, i.e. (A.21) is satisfied, one may apply the discussion above to the universal covers, and obtain a contradiction in the same manner.

■

We note that exactly the same proof can be used to prove also that

|∇ log u|(x) ≤ \frac{K}{t(x)}.  (A.22)

In particular, these results together prove Theorem 3.2.

Note that since K is independent of the domain U, (A.11) holds for t the distance to the event horizon Σ, (provided this is defined), even if Σ is singular. More generally but for the same reasons, (A.11) holds for t the distance to ∂N^{o}, where N^{o} is the maximal domain on which the potential ū is positive, c.f. Theorem 5.1. This is because N^{o} is the Hausdorff limit of an exhaustion of N^{o} by smooth subdomains.

Remark A.4. (i). Similarly, using elliptic regularity associated to the static vacuum equations, one may show in the same way that for any k ≥ 1,

|∇^{k}z|(x) ≤ \frac{C(k)}{t^{2+k}(x)}, |D^{k} log u|(x) ≤ \frac{C(k)}{t^{k}(x)}.  (A.23)

(ii). Under the same hypotheses as Corollary A.3, suppose also that there is an end E ⊂ N such that

u(x) → const. > 0,  (A.24)

as x → ∞ in E. Then essentially the same proof shows that there is a function μ = μ(t) such that

|z|(x) ≤ \frac{μ(t(x))}{t^{2}(x)}, |∇ log u|(x) ≤ \frac{μ(t(x))}{t(x)},  (A.25)

where t(x) is the distance to a fixed base point in E. Namely, if (A.25) does not hold, then one derives the same estimates as (A.12), (A.13), (A.18), with ∞ replaced by some constant c > 0 and concludes the proof as before using the fact that static vacuum solutions with u = const. are flat.

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May, 1998; revision, November, 1998

Department of Mathematics S.U.N.Y. at Stony Brook Stony Brook, N.Y. 11794-3651 E-mail: anderson@math.sunysb.edu