Inverse curvature flows in hyperbolic space

TL;DR

This paper studies inverse curvature flows in hyperbolic space, showing that star-shaped initial hypersurfaces evolve over time to become convex and spherical, with convergence properties proven through geometric analysis.

Contribution

The paper proves long-time existence and convergence of inverse curvature flows in hyperbolic space for star-shaped hypersurfaces, including exponential convexity and spherical limits.

Findings

Flows exist for all time

Hypersurfaces become strongly convex exponentially fast

Rescaled hypersurfaces converge smoothly to a sphere

Abstract

We consider inverse curvature flows in $\Hh$ with star-shaped initial hypersurfaces and prove that the flows exist for all time, and that the leaves converge to infinity, become strongly convex exponentially fast and also more and more totally umbilic. After an appropriate rescaling the leaves converge in $C^\infty$ to a sphere.