Mathematics Department - Geometric Analysis Seminar - Spring 2017

# Geometric Analysis Seminar - Spring 2017

### Organizer(s)

Paul Feehan, Manousos Maridakis, Natasa Sesum

## Tuesday, April 25th

### "Bartnik's quasi-local mass in general relativity"

Time: 3:00 PM
Location: Hill 705
Abstract: In general relativity, the quasi-local mass problem is to find a "good" definition for the amount of mass contained within a compact 3-dimensional spatial region, Omega. Inspired by the classical definition of capacity, Bartnik's definition considers all possible asymptotically flat extensions of Omega (subject to certain natural geometric conditions), and minimizes the total (ADM) mass among such spaces. He conjectured that this infimum is attained. In joint work with Michael Anderson, we show that for a large family of regions, the infimum is not achieved. I will discuss our recent results, including (time-permitting) further results pertaining to Bartnik mass minimizers and static vacuum solutions of Einstein's equations.

## Tuesday, April 11th

### "The Lojasiewicz-Simon gradient inequality and applications to energy discreteness and gradient flows in gauge theory"

Time: 3:00 PM
Location: Hill 705
Abstract: The Lojasiewicz-Simon gradient inequality is a generalization, due to Leon Simon (1983), to analytic or Morse-Bott functionals on Banach manifolds of the finite-dimensional gradient inequality, due to Stanislaw Lojasiewicz (1963), for analytic functions on Euclidean space. We shall discuss several recent generalizations of the Lojasiewicz-Simon gradient inequality and a selection of their applications, such as global existence and convergence of Yang-Mills gradient flow over four-dimensional manifolds and discreteness of the energy spectrum for harmonic maps from Riemann surfaces into analytic Riemannian manifolds. Parts of this talk are joint work with Manousos Maridakis.

## Tuesday, April 4th

### "Convergence of Yamabe flow on manifolds with minimal boundary "

Time: 3:00 PM
Location: Hill 705
Abstract: We will describe the joint work with Sergio Almaraz concerning the Yamabe flow on manifolds with minimal boundary condition. Convergence to a metric with constant scalar curvature and minimal boundary is established in dimensions up to seven and in any dimension if the manifold is spin. Some other flow with dynamic boundary condition will be briefly introduced.

## Tuesday, March 7th

### "Analysis of Velázquez's solution to the mean curvature flow with a type II singularity"

Time: 3:00 PM
Location: Hill 705
Abstract: J.J.L. Velázquez in 1994 used the degree theory to show that there is a perturbation of Simons' cone, starting from which the mean curvature flow develops a type II singularity at the origin. He also showed that under a proper time-dependent rescaling of the solution around the origin, the rescaled flow converges in the C^0 sense to a minimal hypersurface which is tangent to Simons' cone at infi nity. In this talk, we will present that the rescaled flow actually converges locally smoothly to the minimal hypersurface, which appears to be the singularity model of the type II singularity. In addition, we will show that the mean curvature of the solution blows up near the origin at a rate which is smaller than that of the second fundamental form.

This is a joint work with N. Sesum.

## Tuesday, February 21st

### "The extension problem of the mean curvature flow (I)"

Time: 3:00 PM
Location: Hill 705
Abstract: We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in R^3.

This is a joint work with H.Z. Li.

## Tuesday, February 14th

### "Free boundary problems on the Gauss curvature flow"

Time: 3:00 PM
Location: Hill 705
Abstract: We will discuss about the optimal $C^{1,frac{1}{n-1}}$ regularity of the Gauss curvature flow with a flat side and the $C^{infty}$ regularity of its free boundary, namely the boundary of the flat side. We establish estimates by applying the maximum principle for chart independent geometric quantities.

## Tuesday, February 7th

### "Nonlocal Curvature Flows"

Time: 3:00 PM
Location: Hill 705
Abstract: In this talk, some nonlocal planar cuvature flows will be introduced, including the area-preserving flow and the length-preserving one, etc. The locally convex initial curve could be embedded or immersed. Some sharp conditions on immersed initial curves are established to yield the singularity of the flow at a finte time. In addition, the symmetry of curves are found to play an important role in the flow's evolution.

## Tuesday, January 31st

### "Analysis of Velázquez's solution to the mean curvature flow with a type II singularity"

Time: 3:00 PM
Location: Hill 705
Abstract: J.J.L. Velázquez in 1994 used the degree theory to show that there is a perturbation of Simons' cone, starting from which the mean curvature flow develops a type II singularity at the origin. He also showed that under a proper time-dependent rescaling of the solution around the origin, the rescaled flow converges in the C^0 sense to a minimal hypersurface which is tangent to Simons' cone at infi nity. In this talk, we will present that the rescaled flow actually converges locally smoothly to the minimal hypersurface, which appears to be the singularity model of the type II singularity. In addition, we will show that the mean curvature of the solution blows up near the origin at a rate which is smaller than that of the second fundamental form.

This is a joint work with N. Sesum.

## Tuesday, January 24th

Special Geometric Analysis Seminar

### " Surfaces of Low Entropy"

Time: 3:00 PM
Location: Hill 705
Abstract: The entropy of a (hyper)-surface is given by the supremum over all Gaussian integrals with varying centers and scales. It is a geometric invariant that measures complexity of the surface. This quantity is introduced by Colding and Minicozzi to understand the singularity formation of mean curvature flow of surfaces. In this talk, I will discuss several results which show that closed surfaces that have small entropy are simple in various senses.

This is joint with Jacob Bernstein.

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