Quermassintegral preserving curvature flow in Hyperbolic space

TL;DR

This paper studies a curvature flow in hyperbolic space that preserves quermassintegrals, showing that it smooths and converges to a geodesic sphere for h-convex initial hypersurfaces.

Contribution

It introduces a new curvature flow in hyperbolic space that preserves quermassintegrals and proves long-term existence and exponential convergence to spheres.

Findings

Flow preserves quermassintegrals during evolution.

Solutions become strictly h-convex for positive time.

Flow converges exponentially to a geodesic sphere.

Abstract

We consider the quermassintegral preserving flow of closed \emph{h-convex} hypersurfaces in hyperbolic space with the speed given by any positive power of a smooth symmetric, strictly increasing, and homogeneous of degree one function $f$ of the principal curvatures which is inverse concave and has dual $f_*$ approaching zero on the boundary of the positive cone. We prove that if the initial hypersurface is \emph{h-convex}, then the solution of the flow becomes strictly \emph{h-convex} for $t>0$, the flow exists for all time and converges to a geodesic sphere exponentially in the smooth topology.