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

TL;DR

This paper proves uniqueness and symmetry of self-similar solutions to curvature flows in warped product spaces, showing conditions under which solutions are spherical or rotationally symmetric, including in hyperbolic and anti-deSitter-Schwarzschild spaces.

Contribution

It establishes new uniqueness and symmetry results for self-similar solutions in warped product spaces, extending understanding of curvature flows in these geometries.

Findings

Compact solutions with homogeneous degree -1 are slices.

Self-expanders with degree > -1 are also slices under certain conditions.

Non-compact solutions in hyperbolic and anti-deSitter-Schwarzschild spaces are rotationally symmetric.

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 $I \times_{\phi} M^n$, where $M^n$ is a compact homogeneous manifold and $\phi'' \geq 0$, 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 $\phi' \geq 0$. 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.