Contracting Self-similar Solutions of Nonhomogeneous Curvature Flows

A recent article (Li and Lv, J Geom Anal 30:417-447, 2020) 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.

Automated summary

The article discusses the fully nonlinear contraction of convex hypersurfaces by nonhomogeneous functions of curvature, and explores self-similar solutions to curvature flows that are not homogeneous in principle curvatures, revealing situations where curvature-pinched hypersurfaces contracting self-similarly must necessarily be spheres.