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XXth Annual Geometry Festival

SUNY Stony Brook, Stony Brook, NY
April 8-10, 2005

Suggested Research Problems

Nancy Hingston,   Periodic Solutions of Hamilton's Equations on Tori

  1. Let H=H(t,x) be a Hamiltonian of period 1 in the real variable t, defined for points x in a symplectic 2n-ball. Suppose that x=0 is an isolated period 1 orbit of the Hamiltonian flow, and is topologically degenerate, i.e. its iterates are all homologically visible in dimension n. (That is, in a finite dimensional approximation to the free loop space, there is nontrivial local homology in dimension n, using Maslov grading.)
    • Prove, or give a counter example: "Every neighborhood of the origin contains orbits of arbitrarily large minimal (integer) period."
    Note after a change of coordinates, we can assume that H has a strict local minimum at the origin for each fixed t.

  2. Given an isolated closed geodesic γ on a compact Riemannian manifold of dimension n. Assume its iterates are homologically visible (have nontrivial local homology in the free loop space) in a sequence of dimensions exhibiting the fastest (or slowest) possible growth rate in dimension n; that is, the mth iterate is homologically visible in dimension
      D(m) = am ± (n-1)(m-1)
    for some integer a.
    • Prove, or give a counter example: "Every tubular neighborhood of γ contains arbitrarily long prime closed geodesics."
Note in both cases we can prove that there is a sequence of “new” critical values whose limit is the given critical value (suitably renormalized), but it does not obviously follow from the proof that the “new” critical points converge to the given critical point. A “proof” would prove a new theorem or give a new (better) proof of an old theorem. A counterexample would justify our conviction that something interesting is going on: The existence of one topologically degenerate critical point necessitates the existence of infinitely many other critical points, perhaps far away.

Sergiu Klainerman,   Null Hypersurfaces and Curvature Estimates in General Relativity

In my lecture, I mentioned the following result:

Solutions of the Einstein vacuum equations expressed relative to a maximal foliation with second fundamental form κ and lapse ν can be indefinitely extended as long as the L norms of κ and ∇ log ν are bounded.

Problem: As ν satisfies the elliptic equation Δν = ν|κ|2, one should be able to virtually eliminate the condition on ν. We expect that it should suffice to simply require that ν be bounded below.

Bruce Kleiner,   Singular Structure of Mean Curvature Flow

Mean-Curvature Flow:

  • Uniqueness of limit flows: If Λ ⊂ R3×R is a noncompact limit flow, must Λ be either a flat plane, a shrinking round cylinder, or a unique "bowl'' translating soliton?
  • Is there a good regularity theory for level set flows with spherical initial condition? Is there an analog of Perelman's noncollapsing result for general level set flows?
3-Dimensional Ricci Flow:
  • If (M,gt) is a Ricci flow on a compact connected 3-manifold, does the diameter of the time slices (M,gt) remain uniformly bounded up to the first blow-up time?
  • Is the Bryant soliton the unique 1-ended κ-solution (i.e. limit flow in the Ricci world)?
  • Can one show that there is a canonical "Ricci flow with singularities'' with a given initial condition, i.e. a good weak version of Ricci flow?

Franck Pacard,     Blowing Up Kähler Manifolds with Constant Scalar Curvature

Here is a list of potential problems related to my joint papers with C. Arezzo, Blowing up and desinguarizing Kähler orbifolds of constant scalar curvature. and Blowing up Kähler manifolds with constant scalar curvature II. In these two papers we essentially prove that the blow up of Kähler constant scalar curvature metrics at finitely many points carries a Kähler constant scalar curvature metric.

  1. Can the method developed in these two papers be generalized so as to also handle extremal Kähler metrics with non-constant scalar curvature?
  2. Can blow-ups along submanifolds be treated similarly ?
  3. In paper II, a certain analytic condition appears as a sufficient condition for the construction to work. It would be interesting to give a geometric re-interpretation of this condition.

Rahul Pandharipande,   A Topological View of Gromov-Witten Theory

  1. Find a method to efficiently compute the higher genus Gromov-Witten invariants the quintic 3-fold.
  2. Determine the structure of the descendent Gromov-Witten theory of surfaces general type.

Igor Rodniansky ,   Non-Linear Waves and Einstein Geometry

  1. Uniqueness problem in General Relativity: the classical result of Choquet-Bruhat and Geroch (in its refined version due to Hughes-Kato-Marsden) guarantees existence of a maximal Hs Cauchy development for vacuum Einstein equations with arbitrary initial data in Hs for any s>5/2. However, uniqueness (up to a diffeomorphism) of such developments requires the level of regularity Hs with s>7/2. It would be of interest to either remove the discrepancy between the existence and uniqueness results or understand its source. A similar problem can be also investigated for Hs solutions with s>2 constructed by Klainerman-Rodnianski.

  2. Prove the analogue of the "small energy implies regularity" result for the wave-map problem, with target the hyperbolic plane H2 and base-manifold a (2+1)-dimensional Lorentzian "cone" manifold with metric
      -dt2 + t2 σ ,
    where σ a metric of scalar curvature -1 on a compact surface of genus ≥ 2. When the base-manifold is (2+1)-dimensional Minkowski space, the problem was solved by Tao and Krieger. In the case considered here, the base-manifold is spatially compact, but expanding in the direction of positive time. This expansion should replace the dispersive phenomena associated with the wave equation for the problem in the whole space, but as a result one could only hope for a global existence result in the positive time direction. This problem is related to a U(1)-symmetry reduction of the (3+1)-dimensional Einstein vacuum equations to a (2+1)-dimensional problem; cf. e.g. recent work of Choquet-Bruhat and Moncrief.

Yum-Tong Siu,   Methods of Singular Metrics in Algebraic Geometry

  1. Conjecture on Deformation Invariance of Plurigenera.
    Let π : X → Δ be a holomorphic family of compact complex Kähler manifolds over the unit 1-disk Δ ⊂ C. Let Xt-1(t) for t∈ Δ.
    Then dimC Γ (Xt, mKXt) is independent of t, for any positive integer m.

  2. Conjecture on Finite Generation of the Canonical Ring.
    Let X be a compact complex manifold of general type. Let
      R(X,KX)=⊕m=0Γ (X,mKX).
    Then the ring R(X,KX) is finitely generated.

  3. Conjecture on Rationality of Stable Vanishing Order
    (implied by Conjecture 2)
    Let X be a compact complex manifold of general type and let
      s(m)1, … , s(m)qm∈ Γ (X,mKX)
    be a basis over C. Let
      Φ =∑m=1 εm( ∑j=1qm |s(m)j|2)1/m
    where εm is some sequence of positive numbers decreasing fast enough to guarantee convergence of the series. Then all the Lelong numbers of the closed positive (1,1)-current [i/2π] ∂d log Φ are rational numbers.

Katrin Wehrheim,   Floer Theories in Symplectic Topology and Gauge Theory

A mean value inequality with boundary condition
In Energy quantization and mean value inequalities for nonlinear boundary value problems, I developed a mean value inequality which is crucial for the bubbling analysis in several versions of Floer theory, as well as for so-called ε-regularity theorems. This inequality has to do with the standard Laplacian on a Euclidean half-space. However, the analogous interior estimate is well known to hold for curved manifolds. So it would be interesting to

  • Generalize this mean-value inequality to boundaries of curved manifolds.

Lagrangians in the space of connections
Let Y be a compact 3-manifold with boundary ∂ Y=Σ and consider

    ΛY:= { A|Σ s.t. A∈Ω1(Y;su(2)), dA+A ∧ A =0 } ⊂ Ω1(Σ ; su(2)) .
When Y=H is a handle body, I showed in Banach space valued Cauchy-Riemann equations ith totally real boundary conditions that the Lp-closure of ΛH is a Lagrangian Banach submanifold for any p>2.
  • Is the L2-closure of ΛH smooth? (If so, it would be a Lagrangian manifold modelled on a Hilbert space.)
  • Is ΛY smooth for any other 3-manifold Y?

Gauge orbits in the L2-topology.
Consider a smooth connection A ∈ Ω1(Σ ; su(2)) on a Riemann surface Σ.
Let u∈ C(Σ , SU(2)) be a smooth gauge transformation such that

    u*A=u-1Au + u-1d u
is Lp-close to A for some p>2. Then there exists a path
    v:[0,1]→ C(Σ , SU(2))
such that v(0,•)≡ 1, v(1,•)*A=u*A and the Lp-length of the path
    t → v(t,•)*A
is bounded by C |u*A-A|p.
  • Does this also hold for p=2?

Back to the XXth
East Coast Geometry Festival
Web page:  Stony Brook Mathematics
Comments to:  Claude LeBrun
Started: Apr 12, 2005.    Updated: May 23, 2005.
URL: http://www.math.sunysb.edu/geomfest05/abstracts.html