Workshop on Mean Curvature and Regularity

Abstract: In 1963 Brill and Lindquist asked where one might find the apparent horizons of charged black holes in geometrostatic manifolds that arise as time symmetric solutions to vacuum Einstein Maxwell constraint equations. These manifolds are...

Abstract: We define a relative entropy for two expanding solutions to mean curvature flow of hypersurfaces, asymptotic to the same smooth cone at infinity. Adapting work of White and using recent results of Bernstein and Bernstein-Wang, we show that...

Abstract: We will explain how to prove properness of a complete embedded minimal surface in Euclidean three-space, provided that the surface has finite genus and countably many limit ends (and possibly compact boundary).

This is joint work with...

Abstract: I will survey the recent progress on the existence problem for minimal hypersurfaces and then point for some new directions. This is joint work with Fernando Marques.

Abstract: We give some existence and non-existence results for minimal annuli in H2xR whose data at infinity are given.

Abstract: Consider an area minimizing integral current $T$ which has a smooth boundary $\Gamma$ of multiplicity $1$ in some smooth Riemannian ambient manifold $\Sigma$. If the current has codimension $1$ a famous work of Hardt and Simon gives full...

Abstract: In this talk we show that given any regular cone with entropy less than that of round cylinder, all smooth self-expanding solutions of the mean curvature flow that are asymptotic to the cone are in the same isotopy class. This is joint...

Abstract: Let M be a manifold with bounded Ricci curvature and which is noncollapsed, i.e. |Rc|v>0. Together with Wenshuai Jiang we prove that there then exists the apriori L^2 curvature estimate \int |Rm|^2 C and L^1 hessian estimate \int |...

Abstract: The Allen-Cahn equation behaves as a desingularization of the area functional. This allows for a PDE approach to the construction of minimal hypersurfaces in closed Riemannian manifolds. After presenting and overview of the subject, I...

Abstract: The Clifford torus is the simplest nontotally geodesic minimal surface in S^3. It is a product surface, it is helicoidal, and it is a solution obtained by separation of variables. We will show that there are more minimal submanifolds with...

Abstract: I will go over some recent work that I have been involved in on surface geometry in complete locally homogeneous 3-manifolds X. In joint work with Mira, Perez and Ros, we have been able to finish a long term project related to the Hopf...