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

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.