Uniqueness and Symmetry of Self-Similar Solutions of Curvature Flows in Warped Product Spaces

Abstract

In this article, we establish some uniqueness and symmetry results of self-similar solutions to curvature flows by some homogeneous speed functions of principal curvatures in some warped product spaces. In particular, we proved that any compact star-shaped self-similar solution to any parabolic flow with homogeneous degree $-1$ (including the inverse mean curvature flow) in warped product spaces $I{\times }_{\varphi }{M}^n$, where $M^n$ is a compact homogeneous manifold and ${\varphi }^{\prime \prime }\ge 0$, must be a slice. The same result holds for compact self-expanders when the degree of the speed function is greater than $-1$ and with an extra assumption ${\varphi }^{\prime }\ge 0$. Furthermore, we also show that any complete non-compact star-shaped, asymptotically concial expanding self-similar solutions to the flow by positive power of mean curvature in hyperbolic and anti-deSitter-Schwarzschild spaces are rotationally symmetric.

20 pages