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

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 $F=-⟨X,e_{n+1}⟩(\ast )$ 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 $(\ast )$ must be a round sphere.

14 pages