Volume preserving nonhomogeneous Gauss curvature flow in hyperbolic space

TL;DR

This paper studies a volume-preserving curvature flow in hyperbolic space, proving long-term existence, convexity preservation, and exponential convergence to a geodesic sphere for a broad class of curvature-dependent speeds.

Contribution

It establishes the global existence and exponential convergence of the flow in hyperbolic space for nonhomogeneous Gauss curvature speeds, extending previous results to a wider class of functions.

Findings

Flow remains convex for all time

Solutions converge exponentially to geodesic spheres

Decay of Gauss curvature oscillation is proved

Abstract

We consider the volume preserving flow of smooth, closed and convex hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1}$ with speed given by a general nonhomogeneous function of the Gauss curvature. For a large class of speed functions, we prove that the solution of the flow remains convex, exists for all positive time $t\in [0,\infty)$ and converges to a geodesic sphere exponentially as $t\to\infty$ in the smooth topology. A key step is to show the $L^1$ oscillation decay of the Gauss curvature to its average along a subsequence of times going to the infinity, which combined with an argument using the hyperbolic curvature measure theory implies the Hausdorff convergence.