Closed self-similar solutions to flows by negative powers of curvature

TL;DR

This paper investigates closed self-similar solutions to curvature flows with speeds as negative powers of curvature functions in warped product manifolds, proving they are slices of these manifolds, with some solutions not strictly convex.

Contribution

It establishes that such self-similar solutions are slices of warped product manifolds, even when not strictly convex, using a novel auxiliary function for the proof.

Findings

Self-similar solutions are slices of warped product manifolds.

Solutions may not be strictly convex in some cases.

A new auxiliary function is key to the proof.

Abstract

In some warped product manifolds including space forms, we consider closed self-similar solutions to curvature flows whose speeds are negative powers of mean curvature, Gauss curvature and other curvature functions with suitable properties. We prove such self-similar solutions, not necessarily strictly convex for some cases, must be slices of warped product manifolds. A new auxiliary function is the key of the proofs.