Program:

3:00-3:50 PM    Xin Zhou (UC Santa Barbara), `Min-max minimal hypersurfaces with free boundary'

Abstract: I will present a joint work with Martin Li. Minimal surfaces with free boundary are natural critical points of the area functional in compact smooth manifolds with boundary. In this talk, I will describe a general existence theory for minimal surfaces with free boundary. In particular, I will show the existence of a smooth embedded minimal hypersurface with free boundary in any compact smooth Euclidean domain. The minimal surfaces with free boundary were constructed using the min-max method. I will explain the basic ideas behind the min-max theory as well as our new contributions.

4:00-4:50 PM    Vladimir Markovic (Caltech), `Harmonic maps and heat flows on hyperbolic spaces'

Abstract: We prove that any quasi-isometry between hyperbolic manifolds is homotopic to a harmonic quasi-isometry.

A FBMS in the unit Euclidean ball is a critical point of the area functional among all surfaces with boundaries in the unit sphere, the boundary of the ball. The Morse index gives the number of distinct admissible deformations which decrease the area to second order. In this talk, we explain how to compute the index from data of two simpler problems. The first one is the corresponding problem with fixed boundary condition; and the second is associated with the Dirichlet-to-Neumann map for Jacobi fields. We also discuss applications to a conjecture about FBMS with index 4.

We consider a priori estimates of Weyl’s embedding problem of (S^2,g) in general 3-dimensional Riemannian manifold (N^3,\bar g). We establish the mean curvature estimate under natural geometric assumption. Together with a recent work by Li-Wang, we obtain an isometric embedding of (S2,g) in Riemannian manifold. In addition, we reprove Weyl’s isometric embedding theorem in space form under the condition that g \in C^2 with D^2g Dini continuous. 

In this talk, we will discuss recent progress on quasi-local mass in
general relativity focusing on the Wang-Yau quasi-local mass and
discuss how to define other quantities such as angular momentum based
on the ideas and techniques developed in the quasi-local mass. We
will also discuss properties and applications of these newly defined
quantities.

We prove the existence of weak solutions of complex m- Hessian equations on compact Hermitian manifolds for the nonnegative  right hand side belonging to $L^p, p>n/m$ ($n$ is the dimension of the manifold). For smooth, positive data the equation has been recently solved by Sz\'ekelyhidi and Zhang. We also give a stability result for such solutions.