Contracting self-similar solutions of nonhomogeneous curvature flows

TL;DR

This paper investigates self-similar solutions to nonhomogeneous curvature flows of convex hypersurfaces, establishing conditions under which these solutions are necessarily spherical, extending previous results on homogeneous flows.

Contribution

It identifies conditions for self-similar solutions of nonhomogeneous curvature flows to be spherical, expanding understanding beyond homogeneous cases.

Findings

Self-similar solutions are necessarily spheres under certain conditions.

Convex, curvature-pinched hypersurfaces contracting self-similarly are spherical.

Results extend known classifications from homogeneous to nonhomogeneous curvature flows.

Abstract

A recent article by Li and Lv considered fully nonlinear contraction of convex hypersurfaces by certain nonhomogeneous functions of curvature, showing convergence to points in finite time in cases where the speed is a function of a degree-one homogeneous, concave and inverse concave function of the principle curvatures. In this article we consider self-similar solutions to these and related curvature flows that are not homogeneous in the principle curvatures, finding various situations where closed, convex curvature-pinched hypersurfaces contracting self-similarly are necessarily spheres.