2008 DG-MP-PDE Seminar-10

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

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