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

Abstract

By adapting the test functions introduced by Choi–Daskaspoulos (Uniqueness of closed self-similar solutions to the Gauss curvature flow. arXiv:1609.05487, 2016) and Brendle–Choi–Daskaspoulos (Acta Math 219(1):1–16, 2017) and exploring the properties of the k-th elementary symmetric function \(\sigma _{k}\) 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 \(\sigma _{k}^{\alpha }=\langle X,\nu \rangle \), 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 \(F+C=\langle X,\nu \rangle \), where F is a positive homogeneous smooth symmetric function of the principal curvatures and C is a constant.