Volume preserving flows for convex curves and surfaces in the hyperbolic space

Abstract

In this paper, we study the curvature flows of convex curves and surfaces in the hyperbolic space. In the first part, we consider the area preserving and length preserving κα-type curvature flows of smooth closed convex curves in the hyperbolic plane H2 and show that these two types of flows evolve convex curve to a geodesic circle exponentially in C∞ topology. In the second part, we study the volume preserving Gauss curvature flow of smooth closed convex surfaces in H3 and show that the solution remains convex and converges to a geodesic sphere exponentially in C∞ topology.