Gradient estimates for inverse curvature flows in hyperbolic space

TL;DR

This paper establishes optimal gradient estimates and convergence results for hypersurfaces in hyperbolic space evolving under inverse curvature flows, especially for flows driven by powers of curvature functions.

Contribution

It provides new gradient bounds and convergence results for inverse curvature flows in hyperbolic space, including the inverse Gauss curvature flow without additional pinching conditions.

Findings

Optimal gradient estimates for certain inverse curvature flows.

Smooth convergence of rescaled hypersurfaces.

Convergence of inverse Gauss curvature flow without pinching conditions.

Abstract

We prove gradient estimates for hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1},$ expanding by negative powers of a certain class of homogeneous curvature functions. We obtain optimal gradient estimates for hypersurfaces evolving by certain powers $p>1$ of $F^{-1}$ and smooth convergence of the properly rescaled hypersurfaces. In particular, the full convergence result holds for the inverse Gauss curvature flow of surfaces without any further pinching condition besides convexity of the initial hypersurface.