Non-scale-invariant inverse curvature flows in Euclidean space

TL;DR

This paper studies inverse curvature flows of star-shaped hypersurfaces in Euclidean space, showing long-term existence and convergence for certain parameters, and finite-time blow-up for others, with rescaled flows approaching the sphere.

Contribution

It provides a comprehensive analysis of inverse curvature flows for different parameter ranges, including convergence and blow-up behaviors, extending previous results to non-scale-invariant cases.

Findings

Flow exists for all time when 0<p<1 and converges to a sphere after rescaling.

Flow blows up in finite time when p>1 for strictly convex initial hypersurfaces.

Rescaled flows always converge to the unit sphere regardless of p within the studied range.

Abstract

We consider the inverse curvature flows $\dot x=F^{-p}\nu$ of closed star-shaped hypersurfaces in Euclidean space in case $0<p\not=1$ and prove that the flow exists for all time and converges to infinity, if $0<p<1$, while in case $p>1$, the flow blows up in finite time, and where we assume the initial hypersurface to be strictly convex. In both cases the properly rescaled flows converge to the unit sphere.