2008 DG-MP-PDE Seminar-10

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.