In this talk we will discuss the mean curvature flow with surgery and how to extend it to the low entropy, mean convex setting. An application to the topology of low entropy self shrinkers will also be discussed. This is a joint work with Shengwen Wang.

A bounded strictly pseudoconvex domain in C^n, n>1, supports a
unique complete Kahler-Einstein metric determined by the Cheng-Yau solution
of Fefferman's Monge-Ampere equation. The smoothness of the solution of
Fefferman's equation up to the boundary is obstructed by a local CR
invariant of the boundary called the obstruction density. In the case n=2
the obstruction density is especially important, in particular in describing
the logarithmic singularity of the Bergman kernel. For domains in C^2
diffeomorphic to the ball, we motivate and consider the problem of
determining whether the global vanishing of this obstruction implies
biholomorphic equivalence to the unit ball. (This is a strong form of the
Ramadanov Conjecture.)

In this talk, we will discuss the relationship between the Minkowski formula and the quasi-local mass in general relativity, In particular, we will use the Minkowski formula to estimate the quasi-local mass. Combining the estimate and the positive mass theorem, we obtain rigidity theorems which characterize the Euclidean space and the hyperbolic space.

We consider $\sigma_k$-curvature equation with $H_k$-curvature condition on a compact manifold with boundary $(X^{n+1}, M^n, g)$. When restricting to the closure of the positive $k$-cone, this is a fully nonlinear elliptic equation with a fully nonlinear Robin-type boundary condition. We prove a general bifurcation theorem in order to study nonuniqueness of solutions when $2k<n+1$. We explicitly give examples of product manifolds with multiple solutions. It is analogous to Schoen’s example for Yamabe problem on $S^1\times S^{n-1}$. This is joint work with Jeffrey Case and Ana Claudia Moreira.