| 5 January |
Speaker: Yanir Rubinstein (Stanford)
Title: The Cauchy problem for the homogeneous Monge-Ampere equation
Abstract:
The Cauchy problem for the homogeneous real and complex Monge-Ampere equations (HRMA and HCMA) arises naturally as the initial value problem for geodesics in spaces of Kahler metrics. It is expected to be an ``ill-posed problem," and there is no known general method for solving it, contrary to the much-studied Dirichlet problem.
Previously, we proposed a quantization method that we conjectured to solve the Cauchy problem, in its lifespan. We describe our results (joint with S. Zelditch) in this direction, restricting to the real equation (HRMA). This makes contact with classical work of Alexandrov and Pogorelov on flat surfaces in R^3, and relies on tools of convex analysis. It suggests the existence of ``optimal sub-solutions" in the absence of weak solutions.
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| 12 January |
Speaker: Richard Bamler (Princeton)
Title: Stability of hyperbolic manifolds with cusps under Ricci flow
Abstract:
We will show that every hyperbolic manifold of finite volume and
dimension greater or equal to 3 is stable under normalized Ricci flow,
i.e. that every small perturbation of the hyperbolic metric flows back
to the hyperbolic metric again. Here we do not need to make any decay
assumptions on this perturbation. As we will see, the main difficulty in
the proof comes from a weak stability of the cusps which has to do with
the existence of certain cusp deformations. We will overcome this weak
stability by using a new analytical method developed by Koch and Lamm.
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| 19 January |
Speaker: Tom Ilmanen (ETH)
Title: Initial Time Singularities in Mean Curvature Flow
Abstract: TBA
Let M_0 be a closed subset of R^n+1 that is a smooth hypersurface except for a finite number of isolated singular points. Suppose that M_0 is asymptotic to a regular cone near each singular point.
Can we flow M_0 by mean curvature?
Theorem (n < 7): there exists a smooth mean curvature evolution startingat M_0 and defined for a short time 0 < t < epsilon.
Such an initial M_0 might arise as the limit of a smooth mean curvature evolution defined earlier than t=0. Thus, the result allows us to flow through singularities in some cases.
We use a monotonicity formula that complements the monotonicity formula of Huisken. The method applies to other geometric heat flows as well.
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| 26 January |
Speaker: TBA
Title: TBA
Abstract:
TBA
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| 2 February |
Speaker: Ngaiming Mok (part of a series)
Title:
Abstract:
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| 9 February |
Speaker: Weiyong He (Oregon)
Title: The complex Monge-Ampere on compact Kaehler manifolds
Abstract:
We consider the complex Monge-Amp\`ere equation on a
compact K\"ahler manifold $(M, g)$ when the right hand side $F$ has
rather weak regularity. In particular we prove that estimate of
$\t\phi$ and the gradient estimate hold when $F$ is in $W^{1, p_0}$
for any $p_0>2n$. As an application, we show that there exists a
classical solution in $W^{3, p_0}$ for the complex Monge-Amp\`ere
equation when $F$ is in $W^{1, p_0}$.
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| 16 February |
Speaker: Stephan Luckhaus (Leipzig)
Title: Compactness classes of rectifiable varifolds
Abstract:
The talk will mainly be on a result published in JAA
that relaxes the condition that the first variation of the
varifold should be a measure in Allard's rectifiability
result
|
| 23 February |
Speaker: Andras Vasy (Stanford)
Title: From Kerr-de Sitter space-times to asymptotically hyperbolic spaces
Abstract:
I will explain recent results describing the asymptotic behavior of solutions
of wave-type equations on backgrounds such as asymptotically de Sitter-type
spaces and Kerr-de Sitter type spaces (rotating black holes).
These results reduce to the analysis of an operator on a one-dimension lower
space; the key issue is that this operator is elliptic in some regions,
hyperbolic in others, and has radial points for the Hamilton flow over the
interface. This analysis in turn has interesting implications on its own: it
provides a new proof of the analytic continuation of the resolvent of the
Laplacian on asymptotically hyperbolic spaces (previous work of Mazzeo-Melrose
and Guillarmou), and proves non-trapping high energy resolvent estimates, which
had been considered elusive thus far (except under strong assumptions, with
much machinery, in the work of Melrose-Sa Barreto-V.).
|
| 2 March |
Speaker: Zhiqin Lu (Irvine)
Title: Eigenvalues of Collapsing Domains and Drift Laplacian
Abstract:
By introducing a weight function to the Laplace operator, Bakry and Emery
defined the "drift Laplacian to study diffusion processes. Our first main
result is that, given a Bakry-Emery manifold, there is a naturally
associated family of graphs whose eigenvalues converge to the eigenvalues of
the drift Laplacian as the graphs collapse to the manifold. Applications of
this result include a new relationship between Dirichlet eigenvalues of
domains in $\R^n$ and Neumann eigenvalues of domains in $\R^{n+1}$,
variational principles, and a maximum principle. Using our main result and
the maximum principle, we are able to generalize all the results in
Riemannian geometry based on gradient estimates to Bakry-\'Emery manifolds.
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| 9 March |
Speaker: Min-Chun Hong (Brisbane)
Title: The Sacks-Uhlenbeck flow on Riemannian surfaces
Abstract:
In this talk, we study an $\alpha$-flow for the
Sacks-Uhlenbeck functional on Riemannian surfaces and prove that the
limiting map by the $\alpha$-flows is a weak solution to the harmonic map
flow.
By an application of the $\alpha$-flow, we present a simple proof
of an energy identity of a minimizing sequence in each homotopy class.
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| 16 March (Final's Week) |
Speaker: Jeff Viaclovsky (Wisconsin)
Title: Rigidity and stability of Einstein metrics for quadratic curvature
functionals.
Abstract:
I will discuss rigidity (existence or nonexistence of infinitesimal
deformations) and stability (strict local minimization) properties of Einstein
metrics for quadratic curvature functionals on Riemannian manifolds. This is
joint work with Matt Gursky.
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