Self-similar solutions to fully nonlinear curvature flows by high powers of curvature

TL;DR

This paper classifies convex hypersurfaces that shrink self-similarly under certain nonlinear curvature flows driven by high powers of curvature, showing they are necessarily round spheres or slices in specific settings.

Contribution

It proves the uniqueness of round spheres and slices as self-similar solutions for a broad class of fully nonlinear curvature flows with high powers of curvature.

Findings

Round spheres are the only self-similar solutions in Euclidean space.

Slices are the only solutions in the hemisphere for powers ≥ 1.

Self-similar solutions are highly restricted under these flows.

Abstract

In this paper, we investigate closed strictly convex hypersurfaces in $\mathbb{R}^{n+1}$ which shrink self-similarly under a large family of fully nonlinear curvature flows by high powers of curvature. When the speed function is given by powers of a homogeneous of degree $1$ and inverse concave function of the principal curvatures with power greater than $1$, we prove that the only such hypersurfaces are round spheres. We also prove that slices are the only closed strictly convex self-similar solutions to such curvature flows in the hemisphere $\mathbb{S}^{n+1}_{+}$ with power greater than or equal to $1$.