Contraction of convex hypersurfaces by nonhomogeneous functions of curvature

TL;DR

This paper extends previous results on the contraction of convex hypersurfaces by curvature functions, demonstrating convergence to round points under broader conditions and showing exponential convergence in smooth topology.

Contribution

It generalizes earlier work by considering additional cases and establishing exponential convergence to round points with pinching conditions.

Findings

Hypersurfaces converge to round points under new curvature conditions.

Convergence is exponential in the $C^ abla$-topology.

Results hold under initial pinching assumptions.

Abstract

A recent article Li and Lv considered contraction of convex hypersurfaces by certain nonhomogeneous functions of curvature, showing convergence to points in finite time in certain 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 extend the result to various other cases that are analogous to those considered in other earlier work, and we show that in all cases, where sufficient pinching conditions are assumed on the initial hypersurface, then under suitable rescaling the final point is asymptotically round and convergence is exponential in the $C^\infty$-topology.