Self-similar solutions of curvature flows in warped products

Abstract

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 $\mathcal{F}\ge 1$.

19 pages