Volume preserving mean curvature flows in hyperbolic space

Abstract:

The VPMCF is a normalized mean curvature flow with a global forcing term. For a compact hypersurface, it flows while keeping the enclosed volume unchanged. Unlike the MCF, the singularity for such a “global” flow is not well-understood, yet it has potentially many applications in many geometry problems. Based on joint work with Lin (Santa Cruz) and Zhang (Sydney), we consider this VPMCF in hyperbolic space, and prove a dynamical stability theorem near spheres.