Shifted inverse curvature flows in hyperbolic space

TL;DR

This paper introduces a new shifted inverse curvature flow in hyperbolic space, analyzing its behavior for horo-convex hypersurfaces and showing the limiting shape is round for certain parameters, contrasting previous non-shifted results.

Contribution

The paper develops the shifted inverse curvature flow in hyperbolic space and proves the roundness of the limit shape for specific parameter ranges, extending understanding of curvature flows.

Findings

Limiting shape is round for 0<p≤1.

Flow exists maximally and asymptotically.

Contrasts with non-shifted inverse curvature flow results.

Abstract

We introduce the shifted inverse curvature flow in hyperbolic space. This is a family of hypersurfaces in hyperbolic space expanding by $F^{-p}$ with positive power $p$ for a smooth, symmetric, strictly increasing and $1$-homogeneous curvature function $f$ of the shifted principal curvatures with some concavity properties. We study the maximal existence and asymptotical behavior of the flow for horo-convex hypersurfaces. In particular, for $0<p\leq 1$ we show that the limiting shape of the solution is always round as the maximal existence time is approached. This is in contrast to the asymptotical behavior of the (non-shifted) inverse curvature flow, as Hung and Wang [18] constructed a counterexample to show that the limiting shape of inverse curvature flow in hyperbolic space is not necessarily round.