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

TL;DR

This paper studies a volume-preserving curvature flow of convex hypersurfaces in hyperbolic space, proving long-term existence, convexity preservation, and exponential convergence to geodesic spheres, extending understanding of non-local curvature flows.

Contribution

It introduces the first analysis of non-local volume-preserving curvature flows in hyperbolic space with minimal initial convexity assumptions.

Findings

Flow preserves convexity and exists for all time.

Flow converges exponentially to geodesic spheres.

Results apply to arbitrary positive powers of Gauss curvature.

Abstract

We consider the volume preserving flow of smooth, closed and convex hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1} (n\geq 2)$ 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.