Self-similar solutions of $\sigma_k^{\alpha}$-curvature flow

TL;DR

This paper proves that under certain conditions, self-similar solutions to the $\sigma_k^{ ext{alpha}}$-curvature flow are necessarily round spheres, extending to non-homogeneous functions and specific comparison conditions.

Contribution

Introduces a new inequality and establishes conditions under which self-similar solutions are spherical, advancing understanding of curvature flows and their symmetry.

Findings

Self-similar solutions are round spheres under curvature pinching.

Results extend to solutions of $F=-\langle X, e_{n+1}\rangle$ with non-homogeneous $F$.

Comparison with specific functions ensures solutions are spherical.

Abstract

In this paper, employing a new inequality, we show that under certain curvature pinching condition, the strictly convex closed smooth self-similar solution of $\sigma_k^{\alpha}$-flow must be a round sphere. We also obtain a similar result for the solutions of $F=-\langle X, e_{n+1}\rangle \, (*)$ with a non-homogeneous function $F$. At last, we prove that if $F$ can be compared with $\frac{(n-k+1)\sigma_{k-1}}{k\sigma_{k}}$, then a closed strictly $k$-convex solution of $(*)$ must be a round sphere.