We will discuss two types of curvature flows designed to prove isoperimetric type inequalities. The first one is a mean curvature type flow, it was introduced in a previous joint work with Junfang Li in space forms. In a recent joint paper with Junfang Li and Mu-Tao Wang, we consider a similar normalized hypersurface flow in the more general ambient setting of warped product spaces with general base. Under a natural necessary condition, the flow preserves the volume of the bounded domain enclosed by a graphical hypersurface, and monotonically decreases the hypersurface area. Under another condition with is related to the notion of “photon sphere” in general relativity, we establish the regularity and convergence of the flow, thereby solve the isoperimetric problem in warped product spaces. In a similar spirit, I will discuss a inverse mean curvature type flow in hyperbolic space to deal with Alexandrov-Fenchel type isoperimetric inequalities.

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Location: ESB 2012 Pengfei Guan, McGill University