Asymptotic behavior of flows by powers of the Gaussian curvature

TL;DR

This paper studies the long-term behavior of convex hypersurfaces evolving under a curvature-dependent flow, showing convergence to spheres or ellipsoids depending on the parameter, with implications for geometric analysis.

Contribution

It provides new convergence results for a family of curvature flows, including an alternative proof for the affine invariant case, extending understanding of geometric evolution equations.

Findings

Flow converges to a sphere for lpha > 1/(n+2)

Flow converges to an ellipsoid at the affine invariant case lpha = 1/(n+2)

New proof techniques for asymptotic shape analysis of convex hypersurfaces

Abstract

We consider a one-parameter family of strictly convex hypersurfaces in $\mathbb{R}^{n+1}$ moving with speed $- K^\alpha \nu$, where $\nu$ denotes the outward-pointing unit normal vector and $\alpha \geq \frac{1}{n+2}$. For $\alpha > \frac{1}{n+2}$, we show that the flow converges to a round sphere after rescaling. In the affine invariant case $\alpha=\frac{1}{n+2}$, our arguments give an alternative proof of the fact that the flow converges to an ellipsoid after rescaling.