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

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 $1\le k\le n-1$, any strictly convex closed hypersurface in ${\mathbb{R}}^{n+1}$ satisfying ${\sigma }_k^{\alpha }=⟨X,\nu ⟩$, with $\alpha \ge \frac{1}{k}$, must be a round sphere. In fact, we prove a uniqueness result for any strictly convex closed hypersurface in ${\mathbb{R}}^{n+1}$ satisfying $F+C=⟨X,\nu ⟩$, where $F$ is a positive homogeneous smooth symmetric function of the principal curvatures and $C$ is a constant.

23 pages, v2: results improved, references added