Hui Ma

In this paper we study self-similar solutions in warped products satisfying $F-\mathcal{F}=\bar{g}(\lambda (r){\partial }_r,\nu )$, where $\mathcal{F}$ is a nonnegative constant and $F$ is in a class of general curvature functions including powers of mean curvature and Gauss curvature. We show that slices are the only closed strictly convex self-similar solutions in the hemisphere for such $F$. We also obtain a similar uniqueness result in hyperbolic space ${\mathbb{H}}^3$ for Gauss curvature $F$ and $\mathcal{F}\ge 1$.

By adapting the test functions introduced by Choi-Daskaspoulos and Brendle-Choi-Daskaspoulos 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.

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.