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

Abstract

In this paper, we investigate closed strictly convex hypersurfaces in R n+1 which shrink self-similarly under a large family of high order fully nonlinear curvature flows. 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 shrinking spheres. We also prove that slices are the only closed strictly convex self-similar solutions to such curvature flows in the hemisphere S n+1 + with power greater than or equal to 1.