Self-similar solutions of curvature flows in warped products

TL;DR

This paper investigates self-similar solutions to curvature flows in warped product spaces, establishing uniqueness of convex solutions in hemispheres and hyperbolic space for specific curvature functions.

Contribution

It proves the uniqueness of closed convex self-similar solutions in hemispheres and hyperbolic space for a broad class of curvature functions.

Findings

Slices are the only convex solutions in the hemisphere.

Uniqueness of solutions in hyperbolic space for Gauss curvature.

Results apply to curvature functions including powers of mean and Gauss curvature.

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 $\mathcal{F}\geq 1$.