Hopf type theorems for self-similar solutions of curvature flows in $\mathbb{R}^3$

TL;DR

This paper establishes that under certain curvature conditions, the only genus-zero, self-similar solutions to various curvature flows in three-dimensional space are round spheres, extending rigidity results without requiring embeddedness.

Contribution

It proves new rigidity theorems for self-similar solutions of nonlinear curvature flows in ^3, showing they must be round spheres under specific curvature pinching conditions.

Findings

Self-similar solutions are round spheres if genus zero and curvature pinching hold.

Rigidity results apply to flows of powers of mean curvature, harmonic mean curvature, and -Gaussian curvature.

No embeddedness assumption is needed for these results.

Abstract

In this paper we prove rigidity results for two-dimensional, closed, immersed, non-necessarily convex, self-similar solutions of a wide class of fully non-linear parabolic flows in $\mathbb{R}^3$. We show this self-similar solutions are the round spheres centered at the origin provided it has genus zero and satisfies a suitable upper pinching estimate for the Gaussian curvature. As applications, we obtain rigidity results for the round sphere as the only closed, immersed, genus zero, self-similar solution of several well known flows, as the flow of the powers of mean curvature, the harmonic mean curvature flow and the $\alpha$-Gaussian curvature flow for $\alpha\in(0,1/4)$. We remark that our result does not assume any embeddedness condition.