
XXth Annual Geometry Festival
SUNY Stony Brook, Stony Brook, NY
April 810, 2005
Suggested Research Problems

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

 Let H=H(t,x) be a Hamiltonian of period 1 in the real variable
t, defined for points x in a symplectic 2nball. 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.
 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 m^{th}
iterate is homologically visible in
dimension
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 = νκ^{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

MeanCurvature Flow:
 Uniqueness of limit flows: If Λ ⊂ R^{3}×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?
3Dimensional Ricci Flow:
 If (M,g_{t}) is a Ricci flow on a compact connected
3manifold, does the diameter of the time slices (M,g_{t})
remain uniformly bounded up to the first blowup time?
 Is the Bryant soliton
the unique 1ended κsolution (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.
 Can the method developed in these two papers be generalized
so as to also handle extremal
Kähler metrics
with nonconstant scalar curvature?
 Can blowups along submanifolds be treated similarly ?
 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
reinterpretation of this condition.


Rahul Pandharipande,
A Topological View of GromovWitten Theory

 Find a method to efficiently compute the higher genus GromovWitten
invariants the quintic 3fold.
 Determine the structure of the descendent GromovWitten theory of
surfaces general type.


Igor Rodniansky
,
NonLinear Waves and Einstein Geometry


Uniqueness problem in General Relativity: the classical result of
ChoquetBruhat and Geroch (in its refined version due to
HughesKatoMarsden) guarantees existence of a maximal
H^{s} Cauchy development for vacuum Einstein equations
with arbitrary initial data in H^{s} for any s>5/2. However,
uniqueness (up to a diffeomorphism) of such developments
requires the level of regularity H^{s} 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
H^{s} solutions with s>2 constructed by KlainermanRodnianski.

Prove the analogue of the "small energy implies regularity"
result for the wavemap problem, with target the hyperbolic
plane H^{2} and basemanifold a (2+1)dimensional Lorentzian
"cone" manifold with metric
where σ a metric
of scalar curvature 1 on a compact surface of genus ≥ 2.
When the basemanifold is (2+1)dimensional Minkowski
space, the problem was solved by Tao and Krieger.
In the case considered here, the basemanifold 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 ChoquetBruhat and
Moncrief.


YumTong Siu,
Methods of Singular Metrics in Algebraic Geometry

 Conjecture on Deformation Invariance of
Plurigenera.
Let π : X → Δ be a holomorphic family of
compact complex Kähler manifolds over the unit 1disk
Δ ⊂ C. Let X_{t}=π^{1}(t) for t∈ Δ.
Then dim_{C} Γ (X_{t}, mK_{Xt})
is
independent of t, for any positive integer m.
 Conjecture on Finite Generation of the
Canonical Ring.
Let X be a compact complex manifold of general
type. Let R(X,K_{X})=⊕_{m=0}^{∞}Γ
(X,mK_{X}).
Then the ring R(X,K_{X}) is finitely generated.
 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,mK_{X})
be a basis over C. Let
Φ =∑_{m=1}^{∞}
ε_{m}( ∑_{j=1}^{qm}
s^{(m)}_{j}^{2})^{1/m}
where
ε_{m} 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 socalled εregularity
theorems. This inequality has to do with the standard Laplacian on
a Euclidean halfspace. However, the analogous interior estimate
is well known to hold for curved manifolds. So it would be interesting to

Generalize this meanvalue inequality to boundaries of curved manifolds.
Lagrangians in the space of connections
Let Y be a compact 3manifold with boundary ∂ Y=Σ and consider
Λ_{Y}:= { A_{Σ} s.t. A∈Ω^{1}(Y;su(2)), dA+A ∧ A =0 }
⊂ Ω^{1}(Σ ; su(2)) .
When Y=H is a handle body, I showed in Banach space valued CauchyRiemann equations
ith totally real boundary conditions that the
L^{p}closure of Λ_{H}
is a Lagrangian Banach submanifold for any p>2.

Is the L^{2}closure of Λ_{H}
smooth? (If so, it would be a Lagrangian manifold modelled
on a Hilbert space.)

Is Λ_{Y} smooth for any other 3manifold Y?
Gauge orbits in the L^{2}topology.
Consider a smooth connection A ∈ Ω^{1}(Σ ; su(2)) on a Riemann surface Σ.
Let u∈ C^{∞}(Σ , SU(2)) be a smooth gauge transformation such that
is L^{p}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 L^{p}length of the path
is bounded by C u*AA_{p}.

Does this also hold for p=2?


