## 2008 DG-MP-PDE Seminar-10

• Date: 04/29/2008
Lecturer(s):
Pengzi Miao (Monash University)
Location:

University of British Columbia

Topic:

On the size of the outermost minimal surface in a compact 3-manifold with a spherical boundary

Description:

Let \$ M \$ be a compact three dimensional Riemannian manifold with a
non-empty boundary. Suppose \$ S \$ is a boundary component of \$ M \$ such
that its mean curvature vector\uffff points inward. Assume \$ S_H \$ is a
closed minimal surface in \$ M \$ which has the properties that \$ S_H \$
and \$ S \$ bounds a region \$ \Omega \$ in \$ M \$ and there is no other
closed minimal surfaces in \$ \Omega \$. Assuming that \$ M \$ has
nonnegative scalar curvature, we are interested in estimating the area
of \$ S_H \$ from above by the geometry data of \$ S \$. A result of this
type could be viewed as a localized statement of the Riemannian Penrose
Inequality in general relativity. In this talk, we derive such an
inequality under the additional assumption that \$ S \$ is metrically a
round sphere.

Schedule:

3:30pm, WMAX 110