Volume preserving Gauss curvature flow of convex hypersurfaces in the hyperbolic space

Abstract

We consider the volume preserving flow of smooth, closed and convex hypersurfaces in the hyperbolic space ${\mathbb{H}}^{n+1}(n\ge 2)$ with the speed given by arbitrary positive power $\alpha$ of the Gauss curvature. We prove that if the initial hypersurface is convex, then the smooth solution of the flow remains convex and exists for all positive time $t\in [0,\infty )$. Moreover, we apply a result of Kohlmann which characterises the geodesic ball using the hyperbolic curvature measures and an argument of Alexandrov reflection to prove that the flow converges to a geodesic sphere exponentially in the smooth topology. This can be viewed as the first result for non-local type volume preserving curvature flows for hypersurfaces in the hyperbolic space with only convexity required on the initial data.

v2, 29 pages, 3 figures, minor revision in the proof of Lemma 6.6

References in corpus (2)

• Quermassintegral preserving curvature flow in Hyperbolic space
• Volume preserving flow by powers of $k$-th mean curvature