# Uniqueness of closed self-similar solutions to $\sigma_k^{\alpha}$-curvature flow

## TL;DR

This paper proves that strictly convex closed hypersurfaces satisfying certain curvature flow equations are necessarily round spheres, establishing a uniqueness result for solutions involving symmetric functions of principal curvatures.

## Contribution

It introduces new test functions and explores properties of elementary symmetric functions to establish the uniqueness of spherical solutions under specific curvature conditions.

## Key findings

- Any strictly convex closed hypersurface satisfying the given curvature equation is a round sphere.
- The result applies to equations involving $\sigma_k^eta$ with $eta o 1/k$.
- The paper generalizes previous uniqueness results for curvature flows.

## Abstract

By adapting the test functions introduced by Choi-Daskaspoulos \cite{c-d} and Brendle-Choi-Daskaspoulos \cite{b-c-d} and exploring properties of the $k$-th elementary symmetric functions $\sigma_{k}$ intensively, we show that for any fixed $k$ with $1\leq k\leq n-1$, any strictly convex closed hypersurface in $\mathbb{R}^{n+1}$ satisfying $\sigma_{k}^{\alpha}=\langle X,\nu \rangle$, with $\alpha\geq \frac{1}{k}$, must be a round sphere. In fact, we prove a uniqueness result for any strictly convex closed hypersurface in $\mathbb{R}^{n+1}$ satisfying $F+C=\langle X,\nu \rangle$, where $F$ is a positive homogeneous smooth symmetric function of the principal curvatures and $C$ is a constant.