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

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 $1$.

arXiv comment23 pages, revised version, some typos are corrected, to appear in Crelle's Journal