ON STATIONARY VACUUM SOLUTIONS TO THE EINSTEIN EQUATIONS

MICHAEL T. ANDERSON

0. Introduction.

A stationary space-time (M, g) is a 4-manifold M with a smooth Lorentzian metric g, of signature (−, +, +, +), which has a smooth 1-parameter group G ≈ ℝ of isometries whose orbits are time-like curves in M. We assume throughout the paper that M is a chronological space-time, i.e. M admits no closed time-like curves, c.f. §1.1 for further discussion.

Let S be the orbit space of the action G. Then S is a smooth 3-manifold and the projection

π : M → S

is a principle ℝ-bundle, with fiber G. The chronology condition implies that S is Hausdorff and paracompact, c.f. [Ha] for example. The infinitesimal generator of G ≈ ℝ is a time-like Killing vector field X on M, so that

L_{X}g = 0.

The metric g = g_{M} restricted to the horizontal subspaces of T M, i.e. the orthogonal complement of < X > ⊂ T M then induces a Riemannian metric g_{S} on S. Since X is non-vanishing on M, X may be viewed as a time-like coordinate vector field, i.e. X = ∂/∂t, where t is a global time function on M. The time function t gives a global trivialization of the bundle π and so induces a diffeomorphism from M to ℝ × S. The metric g_{M} on M may be then written globally in the form

g_{M} = −u^{2}(dt + θ)^{2} + π^{∗}g_{S},  (0.1)

where θ is a connection 1-form for the ℝ-bundle π and

u^{2} = −⟨X, X⟩ > 0.  (0.2)

The 1-form ξ dual to X is thus given by ξ = −u^{2}(dt + θ). The 1-form θ is uniquely determined by g_{M} and the time function t, but of course changes by an exact 1-form if the trivialization of π is changed. We point out that (M, g_{M}) is geodesically complete as a Lorentzian manifold if and only if (S, g_{S}) is complete as a Riemannian manifold, c.f. Lemma 1.1.

The vacuum Einstein field equations on the space-time (M, g) are

r_{M} = 0,  (0.3)

where r_{M} is the Ricci curvature of (M, g_{M}).

Stationary vacuum space-times are usually considered as the possible final, i.e. time-independent, states of evolution of a physical system, in particular isolated physical systems such as isolated stars or black holes, outside regions of matter. The most important non-trivial example is the Kerr metric, c.f. [W], modeling the time-independent gravitational field outside a rotating star.

Partially supported by NSF Grant DMS 9802722.

It is easy to see from the field equations, c.f.(1.4) below, that there are no non-flat stationary vacuum solutions of the field equations (0.3) whose orbit space is a closed 3-manifold S. Hence, we will always assume that S is an open 3-manifold.

Next, it is natural to consider the class of stationary vacuum space-times which are geodesically complete. In this respect, Lichnerowicz [L, §90] proved that any such solution (M, g) for which the 3-manifold (S, g_{S}) is complete and asymptotically flat is necessarily flat Minkowski space.

The assumption that S is asymptotically flat is very common in general relativity in that such space-times serve as natural models for isolated physical systems, e.g. stars or black holes. The reasoning here is that as one moves further and further away from an isolated gravitational source, the corresponding gravitational field should decay as it does in Newtonian gravity, giving in the limit of infinite distance the empty Minkowski space-time.

However, mathematically the requirement that S is asymptotically flat is a very strong assumption on both the topology and geometry of S outside large compact sets. Further, the reasoning above is not at all rigorous. It presupposes that a geodesically complete stationary solution of the vacuum equations, i.e. a stationary solution without sources, is necessarily empty, and so in particular flat.

Consider the fact that there are geodesically complete, non-stationary vacuum space-times consisting of gravitational waves, c.f. [MTW, §35.9] or [R, §8.8] for example. Again, physically, such spacetimes can be considered as idealized limiting configurations at infinite distance from radiating sources. Similarly, if there does in fact exist a complete non-flat stationary vacuum solution, say (M_{∞}, g_{∞}), then there could well exist models (M, g) for isolated physical systems which are asymptotic to (M_{∞}, g_{∞}) at space-like infinity. For instance, it is not even clear apriori that the curvature of a stationary space-time, vacuum outside a compact source region, should decay anywhere at infinity.

The first main result of this paper is that in fact there are no such non-trivial stationary space-times; this of course places the physical reasoning above on stronger footing.

Theorem 0.1. Let (M, g) be a geodesically complete, chronological, stationary vacuum space-time. Then (M, g) is the flat (i.e. empty) Minkowski space (ℝ^{4}, η), or a quotient of Minkowski space by a discrete group Γ of isometries of ℝ^{3}, commuting with G. In particular, M is diffeomorphic to S × ℝ, dθ = 0 and u = const.

This result, together with Lemma 1.1 below implies that if (M, g) is a non-flat stationary vacuum space-time, then the orbit space S must have a non-empty metric boundary. More precisely, since (S, g_{S}) is Riemannian, let \overline{S} denote the metric, (equivalently the Cauchy), completion of S and let ∂S = \overline{S} \ S. Hence

Σ = ∂S ≠ ∅,  (0.4)

if (M, g) is not flat.

In order to avoid trivial ambiguities, we will only consider maximal stationary quotients S. For example any domain Ω in ℝ^{3} with the flat metric, u a positive constant, and θ = 0 generates a stationary vacuum solution, (namely a domain in Minkowski space). In this case, the metric boundary ∂Ω is artificial, and has no intrinsic relation with the geometry of the solution. The solution obviously extends to a larger domain, i.e. all of Minkowski space. Thus, we only consider maximal solutions (S, g_{S}, u, θ), in the sense that the data (S, g_{S}, u, θ) does not extend to a larger domain (S′, g′_{S}, u′, θ′) ⊃ (S, g_{S}, u, θ) with u′ > 0 on M′. It follows that in any neighborhood of a point q ∈ Σ = ∂S, either the metric g_{S} or the connection 1-form θ degenerates in some way, or u approaches 0 in some way, or both.

Without any further restrictions, the behavior of the data near ∂S can be quite complicated; numerous concrete examples of this can be found among the axi-symmetric stationary, or even axi-symmetric static, i.e. Weyl, solutions; c.f. [A1] for further discussion. In particular, singularities, both of curvature type and of non-curvature type, may form at the boundary. The horizon H = {u = 0}, viewed as a subset of S, may or may not be well-defined in this generality; of course it corresponds to the locus in M where the Killing vector X becomes null. Even when H is well-defined and smooth, in general there may be other, possibly singular, parts to ∂S.

Theorem 0.1 leads to the following apriori estimate on the norm of the curvature of a stationary vacuum solution away from the boundary of S, and on the rate of curvature blow-up on approach to the boundary.

Theorem 0.2. There is a constant K < ∞ such that if (M, g) is any chronological stationary vacuum solution, (not geodesically complete), then

|R_{M}|[x] ≤ K/ρ^{2}[x],  (0.5)

where R_{M} is the curvature tensor of (M, g), [x] is the Killing orbit through x ∈ M and ρ(x) = dist_{g_{S}}([x], ∂S). The constant K is independent of the data (M, g).

Note that Theorem 0.2 implies Theorem 0.1 by letting ρ → ∞. On the other hand, Theorem 0.2 requires Theorem 0.1 for its proof. In particular, this result shows that if ∂S is compact in the completion \overline{S}, then the curvature of (M, g) decays at least quadratically w.r.t. the distance from ∂S.

The contents of the paper are as follows. We discuss some background information and preliminary results in §1, needed for the work to follow. Theorem 0.1 is proved in §2 and Theorem 0.2 is proved in §3.

I would like to thank Piotr Chrusciel and Jim Isenberg for useful discussions, the referee for pointing out some needed clarifications and Grisha Perelman for pointing out an error in a previous version of the paper.

1. Background and Preliminary Results.

§1.1. A stationary space-time (M, g) uniquely determines the orbit data (S, g_{S}, u, Ω) described in §0, where Ω = dθ is the curvature 2-form of the bundle π on S. Conversely, given arbitrary orbit data (S, g_{S}, u, Ω), u > 0, satisfying certain equations, (c.f. (1.3)-(1.6) below), there is a unique stationary space-time (M, g) in the sense of §0, i.e. a chronological space-time with a global isometric ℝ-action with the given orbit data.

Of course, if (M, g) is not chronological, then it will not be uniquely determined by the orbit data. One may for instance take a ℤ-quotient of (M, g), preserving the orbit data. More importantly, if (M, g) is not chronological, then the orbit space S may not be a manifold; even if S is a manifold, it may not be Hausdorff, c.f. [Ha]. Since the arguments to follow are global on S, we require that S is globally well-behaved, which is ensured by the chronology condition. It is not known for instance if Theorem 0.1 is valid without this assumption.

Recall that a space-time (M, g) is geodesically complete if all geodesics in (M, g), parametrized by an affine parameter s, are defined for all s ∈ ℝ. The vertical subspace of T M is the subspace spanned by the Killing field X and the horizontal distribution H is its orthogonal complement in T M, defined by the metric g_{M}.

Lemma 1.1. A stationary space-time (M, g_{M}) is geodesically complete if and only if the orbit space (S, g_{S}) is geodesically complete.

Proof: Suppose (M, g_{M}) is geodesically complete. Let γ be a geodesic in S. Since the projection π : M → S is a principle fiber bundle, with horizontal spaces H ⊂ T M, the geodesic γ may be lifted to a horizontal geodesic γ̄ in (M, g_{M}), with the same parametrization. Since (M, g_{M}) is complete, γ̄ is defined for all values of the parameter, and hence so is γ.

Conversely, suppose (S, g_{S}) is geodesically complete, and hence complete as a metric space. Let γ be a geodesic in M, with affine parameter s and tangent vector T. Then the projection σ = π ∘ γ is a curve in S, whose acceleration is given by

∇_{V} V = \frac{1}{2}κ^{2}∇u^{−2} − \frac{1}{2}κL(V).  (1.1)

Here V = dσ/ds = π_{∗}T, ∇ is the covariant derivative in (S, g_{S}), κ =< X, T >= const and L is the linear map defined by < L(A), B > X = [A, B]^{v} where A, B are horizontal vector fields on M and v is the vertical projection, c.f. [T, Ch.18.3] for example. Conversely, any curve σ satisfying (1.1) lifts to a geodesic in (M, g).

The equations (1.1) form a 2^{nd} order system of ODE w.r.t. the parameter s; note that L(V) is linear in V, while κ is a constant in s, depending linearly on V. By local existence and uniqueness, there exist locally defined solutions σ for arbitrary initial data (x, V (x)) ∈ T S. Since S is complete, it follows that σ exists for all values of s ∈ ℝ. Hence (M, g) is geodesically complete.

Remark 1.2. It is easy to verify that if (M, g) is a stationary, (strongly) globally hyperbolic spacetime, in the sense that (M, g) admits a geodesically complete Cauchy surface L, (w.r.t. the induced metric), then (M, g) is geodesically complete. The converse issue however, i.e. whether a chronological, stationary and geodesically complete space-time is necessarily globally hyperbolic, is not clear to the author, at least without further assumptions on u and θ.

For brevity, we will often say that (M, g_{M}) or (S, g_{S}) is complete instead of geodesically complete.

§1.2. Let ξ = −u^{2}(dt + θ) be the 1-form dual to the Killing vector X, as in §0. The twist potential ω is the 1-form on M defined by

ω = \frac{1}{2} ∗ (ξ ∧ dξ),  (1.2)

It is easily verified that ω is G-invariant, and that it descends to a 1-form ω on the base space S. The form ω represents the obstruction to integrability of the horizontal distribution in T M, and so is related to the curvature 2-form Ω of the connection 1-form θ. In fact, one easily verifies that

2ω = −u^{4} ∗ dθ = −u^{4} ∗ Ω,

on (S, g_{S}).

The vacuum Einstein equations (0.3) on (M, g) are G-invariant, and so also descend to equations on S. The vacuum equations are equivalent to the following equations on (S, g_{S}) :

r = \frac{1}{u}D^{2}u + 2u^{−4}(ω ⊗ ω − |ω|^{2} · g),  (1.3)
Δu = −2u^{−3}|ω|^{2},  (1.4)
divω = 3⟨dlogu, ω⟩,  (1.5)
dω = 0.  (1.6)

Here r = r_{S} is the the Ricci curvature of (S, g_{S}), D^{2}u is Hessian of u on (S, g_{S}), Δu = tr_{g_{S}}D^{2}u and log is the natural logarithm; we refer for instance to [Kr, Ch. 16] for a derivation of these equations, (but note that [Kr] does not use the factor \frac{1}{2} in (1.2)). The equation (1.3) comes from the pure space-like (or horizontal) part of r_{M}, the equation (1.4) from the vertical part of r_{M}, i.e. r_{M}(X, X), while the equations (1.5)-(1.6) come from the mixed directions. The equation (1.6) implies that ω is locally exact, i.e. there exists ϕ, the twist potential, such that

2ω = dϕ  (1.7)

locally. On the universal cover S̃ of S, (1.7) holds globally.

Observe that these equations are invariant under the substitutions

u → λu, ω → λ^{2}ω,  (1.8)

corresponding to ξ → λξ, and θ → λ^{−2}θ.

§1.3. To prove Theorems 0.1 and 0.2, we will need to study sequences of stationary (vacuum) solutions, where all the data (S, g_{S}, u, ω) are allowed to vary. Thus, in effect, we need to understand aspects of the moduli space of stationary solutions. For this, we will frequently use the following two Lemmas, which will be proved together.

Lemma 1.3. (Convergence). Let (Ω_{i}, g_{i}, u_{i}, ω_{i}) represent data for a sequence of solutions to the stationary vacuum equations (0.1). Suppose on the domains (Ω_{i}, g_{i}),

|r_{i}| ≤ Λ, diam Ω_{i} ≤ D, vol Ω_{i} ≥ ν_{o},  (1.9)

and

dist(x_{i}, ∂Ω_{i}) ≥ δ,  (1.10)

for some x_{i} ∈ Ω_{i} and positive constants ν_{o}, Λ, D, δ. Then, for any ε = ε(δ) > 0 sufficiently small, there are domains U_{i} ⊂ Ω_{i}, with ε/2 ≤ dist(∂U_{i}, ∂Ω_{i}) ≤ ε, and x_{i} ∈ U_{i} such that a subsequence of the Riemannian manifolds (U_{i}, g_{i}, x_{i}) converges, in the C^{∞} topology, modulo diffeomorphisms, to a limit manifold (U, g, x), with limit base point x = lim x_{i}.

Further, the potentials u_{i} and 1-forms ω_{i} may be renormalized by scalars λ_{i}, as in (1.8), so that they converge smoothly to limit potential u and 1-form ω. The limit (U, g, x, u, ω) represents a smooth solution to the stationary vacuum equations.

Lemma 1.4. (Collapse). Let (Ω_{i}, g_{i}, u_{i}, ω_{i}) represent data for a sequence of solutions to the stationary vacuum equations (0.1). Suppose on the domains (Ω_{i}, g_{i}),

|r_{i}| ≤ Λ, diam Ω_{i} ≤ D, vol Ω_{i} → 0  (1.11)

and

dist(x_{i}, ∂Ω_{i}) ≥ δ,  (1.12)

for some x_{i} ∈ Ω_{i} and constants Λ, D, δ. Then, for any ε = ε(δ) > 0 sufficiently small, there are domains U_{i} ⊂ Ω_{i}, with ε/2 ≤ dist(∂U_{i}, ∂Ω_{i}) ≤ ε with x_{i} ∈ U_{i}, such that U_{i} is either a Seifert fibered space or a torus bundle over an interval. In both cases, the g_{i}-diameter of any fiber F, (necessarily a circle S^{1} or torus T^{2}), goes to 0 as i → ∞, and π_{1}(F) injects in π_{1}(U_{i}).

Consequently, the universal cover Ũ_{i} of U_{i} does not collapse and hence has a subsequence converging smoothly to a limit (Ũ, g, x), with x = lim x′_{i}, x′_{i} a lift of x_{i} to Ũ_{i}. In addition, the limit (Ũ, g, x) admits a free isometric ℝ-action.

As above, the potentials u_{i} and 1-forms ω_{i}, after possible renormalization by scalars, converge smoothly to limits u and ω. The limit (Ũ, g, x, u, ω) is a smooth solution of the stationary vacuum equations, and all data are invariant under a free isometric ℝ-action on Ũ.

Proofs: The proofs of the first parts of Lemmas 1.3 and 1.4 are essentially immediate consequences of the well-known Cheeger-Gromov theory on convergence and collapse of Riemannian manifolds with bounded curvature, c.f. [CG1,2], [Ka], [A3,§2] for example. We note that we are implicitly using the fact, special to dimension 3, that the full curvature is determined by the Ricci curvature.

More precisely, under the bounds (1.9)-(1.10), one obtains convergence of a subsequence of {g_{i}} to a C^{1,α} limit metric g on the domain U; the convergence is in the C^{1,α′} topology, for any α′ < α < 1. For a clear introduction to this theory, c.f. [P, Ch. 10]. In particular, the bounds (1.9) imply a lower bound on the injectivity radius of every point in U_{i}; this is Cheeger’s lemma, c.f. [C], [P, 10.4.5]

Under the bounds (1.11)-(1.12), the sequence of domains collapses with bounded curvature in the sense that the injectivity radius at every point in U_{i} tends to 0. This implies that the domains U_{i} admit an F-structure, [CG1,2]. In dimension 3, this means that U_{i} is topologically a graph manifold, i.e. a union of Seifert fibered spaces (S^{1} fibrations over a surface) or torus bundles over an interval, glued together along toral boundary components of such, c.f. [Ro, §3]. A result of Fukaya, c.f. [F, Ch.11,12] and references therein, implies that on domains of bounded diameter, i.e. under (1.11)-(1.12), for i sufficiently large, the F-structure may be chosen to be pure, so that U_{i} itself is either a Seifert fibered space or a torus bundle over an interval. The collapse takes place by shrinking the fibers, (circles or tori), to points. From the theory of Seifert fibered spaces, c.f. [O] or [Ro, Thm. 4.3], the fibers inject in π_{1} whenever U_{i} is not covered by S^{3}. But this is necessarily the case here, since U_{i} is an open domain, (c.f. the remark following (0.3). Thus, one may unwrap the collapse by passing to covers, for instance the universal cover, that unwind the fibers. This ability to unwrap collapse on domains of controlled diameter is special to dimension 3.

It remains to show that the convergence is actually smooth (C^{∞}), and that the limit, in either case of Lemma 1.3 or 1.4, is a smooth solution to the stationary vacuum equations. This is done by showing that the equations (1.3)-(1.6) form essentially an elliptic system and using elliptic regularity.

By taking the trace of (1.3) and using (1.4), one derives that

s = −6u^{−4} · |ω|^{2},  (1.13)

where s is the scalar curvature of (S, g_{S}), so that (1.4) is equivalent to

Δu = \frac{s}{3}u.  (1.14)

Since, by hypothesis, the Ricci curvature is uniformly bounded on (Ω_{i}, g_{i}), so is the scalar curvature s_{i}. Now the potential functions u_{i} may be unbounded, or converge to 0, in neighborhoods of the base points x_{i}. Thus, we renormalize u_{i} by setting

ū_{i} = u_{i}/u(x_{i}),  (1.15)

so that ū_{i}(x_{i}) = 1. The equation (1.14) is of course invariant under this renormalization. Moreover, since u_{i} > 0 everywhere, and since the local geometry of (Ω_{i}, g_{i}) is uniformly controlled in C^{1,α} away from ∂Ω_{i}, i.e. within U_{i}, the Harnack inequality, (c.f. [GT, Thm. 8.20]), applied to the elliptic equation (1.14) implies that there is a constant κ > 0, independent of i, such that

κ ≤ \frac{supū_{i}}{infū_{i}} ≤ κ^{−1};  (1.16)

here the sup and inf are taken over U_{i}, or more precisely over an ε/4 thickening of U_{i}. Of course the diameter bound in (1.9) or (1.11) is being used here. It then follows from L^{2} elliptic theory, c.f. [GT, Thm. 9.11], that the functions ū_{i} are uniformly bounded in L^{2,p}(U_{i}), p < ∞. Next, as in (1.15), we renormalize the twist 1-forms ω_{i} by

ω̄_{i} = ω_{i}/(u(x_{i}))^{2},  (1.17)

c.f. (1.8). It then follows from (1.13), (1.15), (1.17) and the uniform L^{∞} bound on s_{i} that the forms ω̄_{i} are uniformly bounded in L^{∞} on U_{i}.

Next, to obtain higher regularity, consider the equations (1.5)-(1.6)

Δϕ_{i} = 3⟨dlogū_{i}, dϕ_{i}⟩,

locally, i.e. in neighborhoods where the twist potential ϕ = ϕ_{i} is defined; (we omit the overbar from the notation for ϕ). We may add a constant to ϕ_{i} and assume ϕ_{i}(x_{i}) = 0. By the bound on ω̄_{i} above, |dϕ_{i}| is uniformly bounded, as is |dlogū_{i}|, so by elliptic regularity, ϕ_{i} is bounded locally in L^{2,p}, and hence ω̄_{i} is uniformly bounded locally in L^{1,p} everywhere in U_{i}. By (1.13) again, this implies s_{i} is bounded in L^{1,p}, and so by elliptic regularity applied to (1.14), ū_{i} is uniformly bounded locally in L^{3,p}. Hence, the right side of (1.3) is bounded in L^{1,p}, and so the Ricci curvature r_{i} is uniformly controlled locally in L^{1,p} everywhere in U_{i}. This implies that the metrics g_{i} are uniformly controlled in L^{3,p} in local harmonic coordinates, c.f. [A3. §3] for example. Hence, by the Sobolev embedding theorem, the sequence {g_{i}} is uniformly bounded in C^{2,α}, α < 1.

This process may now be iterated inductively to give uniform C^{k} control on {g_{i}}, for any k < ∞, away from the boundary, as well as uniform C^{k} control on {ū_{i}} and on {ω̄_{i}}. This proves that the convergence to the limit is in the C^{∞} topology, as well as C^{∞} convergence to limits ū and ω̄. Since the metrics g_{i} are stationary vacuum solutions, it is obvious that the limit (U, g, ū, ω̄) is also.

As an application of these results, we prove the following Lemma, which shows that a given complete stationary vacuum solution gives rise to another one with uniformly bounded curvature.

Lemma 1.5. Let (S, g, u, ω), g = g_{S}, represent data for a complete non-flat stationary vacuum solution. Then there exists another complete non-flat stationary vacuum solution given by data (S′, g′, u′, ω′), g′ = g′_{S′}, obtained as a geometric limit at infinity of (S, g), which has uniformly bounded curvature, i.e.

|r_{g′}| ≤ 1 and |r_{g′}|(y) > 0,  (1.18)

for some y ∈ S′.

Proof: We may assume that (S, g) itself has unbounded curvature, for otherwise there is nothing to prove since (1.18) can then be obtained by a fixed rescaling of (S, g) if necessary. Let {x_{i}} be a sequence in S such that

|r|(x_{i}) → ∞, as i → ∞.  (1.19)

Let B_{i} = B_{x_{i}}(1) and let d_{i}(x) = dist(x_{i}, ∂B_{i}). Consider the scale-invariant ratio (d_{i}^{2}·|r|)(x), for x ∈ B_{i}, and choose points y_{i} ∈ B_{i} realizing the maximum value of (d_{i}^{2} · |r|)(x) on B_{i}. Since (d_{i}^{2} · |r|)(x) is 0 on ∂B_{i}, y_{i} is in the interior of B_{i}. By (1.19), we have

d_{i}^{2}(y_{i}) · |r|(y_{i}) → ∞, as i → ∞  (1.20)

and so in particular |r|(y_{i}) → ∞.

Now consider the pointed rescaled sequence (B_{i}, g_{i}, y_{i}), where

g_{i} = |r|(y_{i}) · g.

By construction, |r_{i}|(y_{i}) = 1, where r_{i} is the Ricci curvature of g_{i}. This, together with (1.20) and its scale-invariance, implies that δ_{i}(y_{i}) ≡ dist_{g_{i}}(y_{i}, ∂B_{i}) → ∞. Further, by the maximality property of y_{i},

|r_{i}|(x) ≤ |r_{i}|(y_{i}) · \frac{δ_{i}(x)}{δ_{i}(y_{i})} = \frac{δ_{i}(x)}{δ_{i}(y_{i})}.  (1.21)

It follows from (1.20) that |r_{i}|(x) ≤ 2, at all points x of uniformly bounded g_{i}-distance to y_{i}, (for i sufficiently large, depending on dist_{g_{i}}(x, y_{i})).

If the pointed sequence (B_{i}, g_{i}, y_{i}), (or a subsequence), is not collapsing at y_{i}, i.e. the volume of the unit g_{i}-ball at y_{i} is bounded below as i → ∞, then by Lemma 1.3, {(B_{i}, g_{i}, y_{i})} has a subsequence converging, smoothly and uniformly on compact subsets, to a limit (U′, g′, y), y = lim y_{i}. The limit is a complete stationary vacuum solution, (since δ_{i}(y_{i}) → ∞), and by the smooth convergence, |r_{g′}| ≤ 2 everywhere and |r_{g′}(y)| = 1, where y = limy_{i}. A further bounded rescaling then gives (1.18). The limit potential u and twist form ω are obtained as in Lemma 1.3.

On the other hand, suppose this sequence is collapsing at y_{i}, so that the volume of the unit g_{i}-ball at y_{i} converges to 0, (in some subsequence). Then by Lemmas 1.3 and 1.4, it is collapsing everywhere within g_{i}-bounded distance to y_{i}, i.e. within (B_{y_{i}}(R), g_{i}), for any fixed R < ∞. For any such R, if i is sufficiently large, there are domains U_{i}(R) ⊂ B_{y_{i}}(R), with ∂U_{i}(R) near ∂B_{y_{i}}(R) w.r.t. g_{i}, which are highly collapsed along an injective Seifert fibered structure or torus bundle structure on U_{i}(R). Hence the universal cover (Ũ_{i}(R), g̃_{i}) is not collapsing. For any sequence R_{j} → ∞, there is then a suitable diagonal subsequence U_{i_{j}} such that the covers Ũ_{i_{j}} converge smoothly, as above, to a complete stationary vacuum solution; again a bounded rescaling then gives (1.18).

2. Proof of Theorem 0.1.

Let (M, g_{M}) be a complete stationary vacuum solution. As above in §1.2 and §1.3, we will work exclusively on the 3-manifold quotient S, with data u, ω and g satisfying the field equations (1.3)-(1.6). By passing to the universal cover, we may and will assume for this section that S is simply connected.

It is very useful to rewrite the metric g_{M} in (0.1) in the form

g_{M} = −u^{2}(dt + θ)^{2} + \frac{1}{u^{2}} ḡ_{S},  (2.1)

where ḡ_{S} is the conformally equivalent metric

ḡ_{S} = u^{2} · g_{S}  (2.2)

on S. Using standard formulas for behavior under conformal changes, c.f. [B, Ch. 1J], w.r.t this metric the field equations (1.3)-(1.5) are equivalent to:

r̄ = 2(dlogu)^{2} + 2u^{−4}(ω)^{2},  (2.3)
Δ̄logu = −2u^{−4}|ω|^{2},  (2.4)
divω = 4⟨dlogu, ω⟩,  (2.5)

c.f. also [Kr, Ch. 16]. All metric quantities in (2.3)-(2.5) are w.r.t. the ḡ = ḡ_{S} metric.

There are two reasons for preferring ḡ to g = g_{S}. First, it is apparent from (2.3) that

r̄ ≥ 0,  (2.6)

so that (S, ḡ) has non-negative Ricci curvature. Second, the field equations (2.3)-(2.5) are exactly the Euler-Lagrange equations for the functional

S_{eff} = ∫_{S}(s − \frac{1}{2}(\frac{|du^{2}|^{2} + |dϕ|^{2}}{u^{4}} ))dV.  (2.7)

Here we are using the fact that S is simply connected, so that the relation (1.7) holds globally on S. This functional is the Einstein-Hilbert functional on G-invariant metrics on M, dimensionally reduced to a functional on data (ḡ, u, ϕ) on S, when g_{M} is expressed in the form (2.1). It corresponds to a coupling of 3-dimensional gravity to the energy (or σ-model) of the mapping E = (ϕ, u^{2}) from S to the hyperbolic plane. The mapping E is called the Ernst potential and the Euler-Lagrange equations (2.3)-(2.5) imply that

E : (S, ḡ_{S}) → (H^{2}(−1), g_{−1})  (2.8)

is a harmonic map. Here H^{2}(−1) is the hyperbolic plane, given as the upper half-plane (ℝ^{2})^{+} = {(x, y) : y > 0}, with metric

g_{−1} = \frac{dx^{2} + dy^{2}}{y^{2}} .  (2.9)

We refer for instance to [H1] or [H2] for further details and discussion on S_{eff}.

From the equation (2.3), we see that

r̄ = \frac{1}{2}E^{∗}(g_{−1}).  (2.10)

In particular, the energy density of e(E) of E, given by

e(E) = \frac{1}{2}|E_{∗}|^{2}

satisfies

s̄ = e(E) = \frac{1}{2}trḡE^{∗}(g_{−1}).  (2.11)

For clarity, we break the proof up at this stage into two steps.

Step I. Assume the metric (S, ḡ_{S}) is complete.

The space (S, ḡ_{S}) may or may not have uniformly bounded curvature, i.e. possibly after a bounded rescaling,

|r̄| ≤ 1,  (2.12)

everywhere on S, where the norm is taken w.r.t. ḡ_{S}. If (2.12) holds, then the arguments below are applied to (S, ḡ_{S}). If instead the curvature of (S, ḡ_{S}) is unbounded, (and hence (S, ḡ_{S}) is not flat), we apply Lemma 1.5 to obtain a new non-flat stationary space-time (S′, ḡ_{S′}, u′, ω′) satisfying (2.12). The arguments below are then applied to (S′, ḡ_{S′}).

With this understood, we drop the prime from the notation and assume that (S, ḡ_{S}) satisfies (2.12).

We now apply the well-known Bochner formula, c.f. [EL, (3.12)], to the harmonic Ernst map E, to obtain

Δ̄e(E) = | ¯∇DE|^{2} + ⟨r_{M}, E^{∗}(g_{−1})⟩ − ∑_{i,j=1}^{3}(E^{∗}R_{−1})(e_{i}, e_{j}, e_{j}, e_{i}).  (2.13)

Here the sign of the curvature tensor for the last term is such that R_{−1}(X, Y, Y, X) is the sectional curvature of g_{−1} for an orthonormal pair (X, Y). We claim that the last two terms in (2.13) are given by

⟨r̄, E^{∗}(g_{−1})⟩ = 2|r̄|^{2},  (2.14)
− (E^{∗}R_{g_{−1}})(e_{i}, e_{j}, e_{j}, e_{i}) = 4(s̄^{2} − |r̄|^{2}) ≥ 0.  (2.15)

The equation (2.14) follows immediately from (2.10). For (2.15), using the fact that g_{−1} is of constant sectional curvature −1, we have −(E^{∗}R_{−1})(e_{i}, e_{j}, e_{j}, e_{i}) = g_{−1}(E_{∗}e_{i}, E_{∗}e_{i}) · g_{−1}(E_{∗}e_{j}, E_{∗}e_{j}) − g_{−1}(E_{∗}e_{i}, E_{∗}e_{j})^{2}. Choosing {e_{i}} to be an orthonormal basis in (S, ḡ_{S}) diagonalizing the Ricci curvature r̄, and using (2.10), gives (2.15).

In particular, the equations (2.13)-(2.15) show that the energy density e(E) is a subharmonic function on (S, ḡ_{S}).

Since (2.12) holds on (S, ḡ_{S}), (2.11) implies that e(E) is uniformly bounded above on (S, ḡ_{S}). Thus, let {x_{i}} be a maximizing sequence for e(F), i.e.

e(F)(x_{i}) → sup e(F) < ∞.  (2.16)

Since the curvature of (S, ḡ_{S}) is bounded, and this space is complete, it follows from elementary properties of the Laplacian that

Δe(F)(x_{i}) ≤ ε_{i},

where ε_{i} → 0, as i → ∞. However, (2.13)-(2.15) then imply that

|r̄|^{2}(x_{i}) ≤ ε_{i} → 0.

This of course forces e(E)(x_{i}) = s̄(x_{i}) → 0. Since x_{i} is a maximizing sequence, this is only possible if

e(E) ≡ 0,

i.e. E is a constant map. This means that u = const > 0, ϕ = const, and hence (M, g) is flat. Thus (M, g) is Minkowski space, (since S is simply connected).

Observe that this argument now implies that the passage to the geometric limit (S′, ḡ_{S′}) at the beginning of Step I was not in fact necessary.

Step II. We now remove the assumption that ḡ is complete, by transfering the estimates above back to the complete manifold (S, g_{S}).

Exactly as in the beginning of Step I however, since (S, g_{S}) is complete, if necessary we use Lemma 1.5 first to pass to a non-flat geometric limit (S′, g_{S′}) with uniformly bounded g′-curvature, i.e. satisfying (1.18). As before, we drop the prime from the notation below.

Since ḡ_{S} = u^{2}g_{S}, we have the following relation between the Laplacians of g_{S} and ḡ_{S}, c.f. [B, Ch. 1J] for example:

Δ̄f = u^{−2}Δf + u^{−3}⟨du, df⟩,

for any function f, where metric quantities on the right are w.r.t. g_{S}. Setting f = s̄ then gives

Δs̄ = u^{2}Δ̄s̄ − ⟨dlogu, ds̄⟩.  (2.17)

Now the function s̄ may well be an unbounded function on (S, g_{S}); (in fact the unboundedness may cause the incompleteness of ḡ_{S}). However, in terms of the metric g, we have

s̄ = u^{−2}(2|dlogu|^{2} + \frac{1}{2}u^{−4}|dϕ|^{2}) ≡ u^{−2} · h,  (2.18)

where the last inequality defines h and the norms on the right are w.r.t. g_{S}. This follows by taking the trace of (2.3).

Since the curvature of g_{S} is uniformly bounded, i.e. (1.18) holds, the same arguments as in the proof of Lemma 1.3-1.4 imply that

|dlogu|^{2} + u^{−4}|dϕ|^{2} ≤ C,  (2.19)

for some C < ∞. The estimate (2.19) can also be deduced directly from (1.13) and (1.3)-(1.7). Hence, h is uniformly bounded above on (S, g_{S}).

Returning to (2.17), we then have

Δs̄ = Δu^{−2}h = u^{−2}Δh + hΔu^{−2} + 2⟨du^{−2}, dh⟩.  (2.20)

Now

Δu^{−2} = −2u^{−3}Δu + 6u^{−4}|du|^{2} = u^{−6}|dϕ|^{2} + 6u^{−4}|du|^{2},  (2.21)

where the last equality uses (1.4) and (1.7). Hence, combining (2.20)-(2.21), we obtain

Δh = u^{2}Δs̄ − (u^{−4}|dϕ|^{2} + 6u^{−2}|du|^{2})h − 2u^{2}⟨du^{−2}, dh⟩.

Substituting (2.17) gives

Δh = u^{4}Δ̄s̄ − (u^{−4}|dϕ|^{2} + 6u^{−2}|du|^{2})h − 2u^{2}⟨du^{−2}, dh⟩ − u^{2}⟨dlogu, ds̄⟩.  (2.22)

Since s̄ = u^{−2} · h, ds̄ = −2u^{−3}hdu + u^{−2}dh, and so (2.22) becomes

Δh = u^{4}Δ̄s̄ − (u^{−4}|dϕ|^{2} + 6u^{−2}|du|^{2})h + 4⟨dlogu, dh⟩ + 2u^{−2}h|du|^{2} − ⟨dlogu, dh⟩,

i.e.

Δh = u^{4}Δ̄s̄ − (u^{−4}|dϕ|^{2} + 4u^{−2}|du|^{2})h + 3⟨dlogu, dh⟩.

By (2.18) again, the middle term on the right above equals −2h^{2} = −2u^{4}s̄^{2}. Hence, we have

Δh − 3⟨dlogu, dh⟩ = u^{4}Δ̄s̄ − 2u^{4}s̄^{2}.  (2.23)

On the other hand, from the Bochner formula (2.13) and (2.14)-(2.15), we have

Δ̄s̄ = | ¯∇DE|^{2} + 2|r̄|^{2} + 4(s̄^{2} − |r̄|^{2}),

where all quantities are w.r.t. the ḡ metric. Substituting this in (2.23) then gives

Δh − 3⟨dlogu, dh⟩ = u^{4}| ¯∇DE|^{2} + 2u^{4}(s̄^{2} − |r̄|^{2}) ≥ 0,  (2.24)

where the terms on the left are in the g metric while those on the right are in the ḡ metric.

We now basically repeat the argument above in Step I to prove that

h ≡ 0.  (2.25)

Thus, recalling from (2.19) that h is bounded on (S, g_{S}), let {x_{i}} be a maximizing sequence for h. It follows as before that Δh(x_{i}) ≤ ϵ_{i}, |dh|(x_{i}) ≤ ϵ_{i} while |dlogu|(x_{i}) remains uniformly bounded.

To prove (2.25), it is most convienient to pass to the limit of the pointed sequence (S, g_{S}, x_{i}) by use of Lemmas 1.3-1.4. Thus, a subsequence of {(S, g_{S}, x_{i})} converges smoothly, (passing to covers if necessary in the case of collapse), to a complete stationary vacuum solution (S_{∞}, g_{∞}, x_{∞}). Here the limit potentials u_{∞} and ϕ_{∞} are limits of the renormalized potentials u_{i} = u/u(x_{i}), ϕ_{i} = ϕ/u(x_{i})^{2}. Observe that h and dlogu are invariant under such renormalizations, as is the right side of (2.24) under the changes u → u_{i}, ḡ_{S} → ḡ_{i} = u_{i}^{2} · g_{S}.

It follows from these estimates and (2.24), together with the maximum principle, that the limit (S_{∞}, g_{∞}, x_{∞}, u_{∞}, ϕ_{∞}) satisfies

h ≡ h_{∞} = const, | ¯∇DE| = 0, |r̄|^{2} − s̄^{2} = 0,  (2.26)

where ḡ_{∞} = u_{∞}^{2} · g_{∞} and

h_{∞} = sup_{S}h.  (2.27)

To see that h_{∞} = 0, (2.26) and (2.10) imply that ¯∇r̄ = 0, i.e. the Ricci curvature r̄_{∞} of ḡ_{∞} is parallel. By the Bianchi identity this implies that the scalar curvature s̄_{∞} of ḡ_{∞} is constant. Since h = h_{∞} is constant, (2.18) shows that u_{∞} is also constant on (S_{∞}, g_{∞}). Hence by (2.4) on (S_{∞}, g_{∞}), it follows that dϕ_{∞} = 0. By the definition of h in (2.18), this of course gives h_{∞} ≡ 0, which by (2.27) gives (2.25).

The equation (2.25) means that u is a constant function and ω = 0, so that dθ = 0. It follows that (S, g_{S}) and (M, g_{M}) are both flat, which proves the result.

3. Proof of Theorem 0.2.

The following result gives Theorem 0.2 essentially as an immediate corollary. The proof is a standard consequence of the global result in Theorem 0.1, together with the control on moduli of stationary vacuum solutions given in Lemmas 1.3 and 1.4.

Theorem 3.1. Let (M, g_{M}) be a stationary vacuum solution, with orbit data (S, g_{S}, u, θ), and U ⊂⊂ S a domain with smooth boundary, so that u > 0 on \overline{U}. Then there is an (absolute) constant K < ∞, independent of (M, g_{M}) and U, such that for all x ∈ U,

|r_{S}|(x) ≤ \frac{K}{ρ(x)^{2}} ,  (3.1)

where ρ(x) = dist_{g_{S}}(x, ∂U).

Proof: The proof is by contradiction. Thus, assume that (3.1) does not hold. Then there are stationary vacuum solutions (M_{i}, g_{M_{i}}), with orbit data (S_{i}, g_{S_{i}}, u_{i}, ω_{i}), smooth domains U_{i} ⊂⊂ S_{i} on which u_{i} > 0 and points x_{i} ∈ U_{i} such that

ρ^{2}(x_{i})|r_{i}|(x_{i}) → ∞, as i → ∞.  (3.2)

Let ρ_{i} = ρ(x_{i}). Since it may not be possible to choose the points x_{i} so that they maximize |r_{i}| (over large domains), we shift the base points x_{i} as follows; compare with the proof of Lemma 1.5. Choose t_{i} ∈ [0, ρ_{i}) such that

t_{i}^{2}sup_{B_{x_{i}}(ρ_{i}−t_{i})}|r_{i}| = sup_{t∈[0,ρ_{i})}t^{2} · sup_{B_{x_{i}}(ρ_{i}−t)}|r_{i}| → ∞, as i → ∞,  (3.3)

where the last estimate follows from (3.2), (set t = ρ_{i}). Let y_{i} ∈ B_{x_{i}}(ρ_{i} − t_{i}) be points such that

|r_{i}|(y_{i}) = sup_{B_{x_{i}}(ρ_{i}−t_{i})}|r_{i}|.  (3.4)

Further, setting t = t_{i}(1 − \frac{1}{k}), k > 1, in (3.3), one obtains the estimate

t_{i}^{2}|r_{i}|(y_{i}) ≥ t_{i}^{2}(1 − \frac{1}{k})^{2} · sup_{B_{x_{i}}(ρ_{i}−t_{i}(1− \frac{1}{k}))}|r_{i}| ≥ t_{i}^{2}(1 − \frac{1}{k})^{2} · sup_{B_{y_{i}}(t_{i}/k)}|r_{i}|,  (3.5)

so that

sup_{B_{y_{i}}(t_{i}/k)}|r_{i}| ≤ (1 − \frac{1}{k})^{−2}|r_{i}|(y_{i}),  (3.6)

Now rescale or blow-up the metric so that |r̃_{i}|(y_{i}) = 1 by setting g̃_{i} = |r_{i}|(y_{i}) · g, and consider the pointed sequence (U_{i}, g̃_{i}, y_{i}). We have

|r̃_{i}|(y_{i}) = 1,  (3.7)

and by (3.3) and scale invariance,

distg̃_{i}(y_{i}, ∂U_{i}) → ∞, as i → ∞.  (3.8)

Also, (compare with (1.21)), it follows from (3.6) that

|r̃_{i}|(x) ≤ C(distg̃_{i}(x, y_{i})).  (3.9)

We also normalize u by setting

ũ_{i}(x) = \frac{u(x)}{u(y_{i})},  (3.10)

and note that ũ_{i} > 0 on U_{i}.

We may now apply Lemmas 1.3 and 1.4, exactly as in the proof of Lemma 1.5 to conclude that a subsequence of the pointed sequence (U_{i}, g̃_{i}, ũ_{i}, ω̃_{i}, y_{i}) converges in the C^{∞} topology on compact subsets, to a limit stationary vacuum solution (U_{∞}, g̃_{∞}, ũ_{∞}, ω̃_{∞}, y), which is complete and satisfies ũ_{∞} > 0 everywhere. Here, one must pass to the universal cover in case of collapse, as in Lemma 1.4, and the potential ũ_{i} and 1-form ω̃_{i} are normalized so that ũ_{i}(y_{i}) = 1 and |ω̃_{i}(y_{i})| is bounded.

By Theorem 0.1, g̃_{∞} must be flat, ũ_{∞} constant and dω̃_{∞} = 0. However, the smooth convergence of the sequence (U_{i}, g̃_{i}) guarantees that the equality (3.7) passes to the limit, contradicting the fact that g̃_{∞} is flat.

As in the proof of Lemmas 1.3 and 1.4, it follows from (3.1) that

|dlogu|(x) ≤ \frac{K}{ρ(x)},  (3.11)

and

u^{−2}|ω|(x) ≤ \frac{K}{ρ(x)}.  (3.12)

Combining the estimates (3.1) and (3.11)-(3.12), one obtains the same bound on the full curvature tensor R_{M} of (M, g).

Note that since K is independent of the domain U, (3.1) holds for ρ the distance to the boundary Σ of S, even if Σ is singular. To see this, just apply Theorem 3.1 to a smooth exhaustion U_{j} of S, with ∂U_{j} converging to ∂S in the Hausdorff metric on subsets of (S, g_{S}). In particular, these results together prove Theorem 0.2.

We note that elliptic regularity further implies that, for any j ≥ 1,

|∇^{j}R_{M}|(x) ≤ \frac{K(j)}{ρ^{2+j}(x)}, |∇^{j}logu|(x) ≤ \frac{K(j)}{t^{j}(x)}.  (3.13)

Theorem 0.2, when combined with Lemmas 1.3 and 1.4, shows that the moduli space of stationary vacuum solutions is apriori well-controlled away from the boundary Σ = ∂S. Thus, away from the boundary, sequences of such metrics either have a smoothly convergent subsequence, or they collapse, in which case the universal covers have a convergent subsequence.

Theorems 0.1 and 0.2 give new proofs of similar results for static vacuum solutions in [An2, Thm. 3.2]. Similarly, in work to follow, we plan to consider generalizations of the results on the asymptotic structure of static vacuum space-times in [A1] to stationary space-times as well as consider the Riemannian analogues of these questions.

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October 1999/March 2000 Department of Mathematics S.U.N.Y. at Stony Brook Stony Brook, N.Y. 11794-3651 anderson@math.sunysb.edu