Volume preserving nonhomogeneous Gauss curvature flow in hyperbolic space

TL;DR

The paper analyzes a volume-preserving, nonhomogeneous Gauss curvature flow for smooth, closed convex hypersurfaces in hyperbolic space $\mathbb{H}^{n+1}$. By establishing time-dependent two-sided curvature bounds, monotonicity of quermassintegrals, and a curvature-measure–theoretic framework, the authors prove global existence and exponential convergence to a geodesic sphere of the same enclosed volume, together with subsequential Hausdorff convergence. A key innovation is a direct curvature-control strategy using an auxiliary function to obtain a uniform lower bound on principal curvatures, avoiding projection methods, and combining curvature measure theory to upgrade subsequential convergence to full smooth convergence. The results extend nonhomogeneous volume-preserving curvature flows in hyperbolic space beyond homogeneous cases, with implications for geometric analysis and isoperimetric-type problems in negatively curved spaces.

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.

Theorems & Definitions (21)

• Remark 1.2
• Definition 1.3
• Theorem 1.4
• Remark 1.5
• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 3.1
• Lemma 3.2
• Proposition 3.3