## 2008 DG-MP-PDE Seminar-10

- Date: 04/29/2008

University of British Columbia

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

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.

3:30pm, WMAX 110