Colloquium

The isoperimetric problem in general manifolds and a mean curvature type flow

 Professor Pengfei

University of McGill, Canada

 Abstract: The isoperimetric problem has been attracting great deal attention for centuries. We all know the isoperimetric inequality in Euclidean space, where the round balls optimize the isometric ratio. The same is also true in hyperbolic space and elliptic space. These three models are so called the space forms a special type of warped product space. There are other important spaces are of warped type, e.g., the anti-de Sitter-Schwarzchild space in general relativity. We consider isoperimentric problem in this type of spaces, via a mean curvature type flow. Such flow was introduced in a previous work to handle the isoperimetric problem in space forms. We study a similar normalized hypersurface flow in the more general ambient setting of warped product spaces with general base. This flow preserves the volume of the bounded domain enclosed by the hypersurface, and monotonically decreases the hypersurface area under a proper setting. As a consequence, the isoperimetric problem can be solved for such domains. We will also discuss the cases where isoperimetric problem is not realized by the standard balls, which is related to a stability condition and rigidity of isometric embeddings. This is a join work with Junfang Li and Mu-Tao Wang. The talk is for general audience.