Non-collapsing in fully nonlinear curvature flows

TL;DR

This paper establishes non-collapsing estimates for embedded hypersurfaces evolving under fully nonlinear curvature flows, showing that certain curvature functions of interior or exterior spheres are subsolutions of the linearized flow, depending on the convexity or concavity of the speed.

Contribution

It proves new non-collapsing estimates for fully nonlinear curvature flows, extending previous results to a broader class of flows with concave or convex speeds.

Findings

Curvature of the largest interior sphere is a subsolution for concave speeds.

Curvature of the largest exterior sphere is a subsolution for convex speeds.

For positive speeds with concavity, the sphere curvature is bounded by a multiple of the speed.

Abstract

We consider embedded hypersurfaces evolving by fully nonlinear flows in which the normal speed of motion is a homogeneous degree one, concave or convex function of the principal curvatures, and prove a non-collapsing estimate: Precisely, the function which gives the curvature of the largest interior sphere touching the hypersurface at each point is a subsolution of the linearized flow equation if the speed is concave. If the speed is convex then there is an analogous statement for exterior spheres. In particular, if the hypersurface moves with positive speed and the speed is concave in the principal curvatures, then the curvature of the largest touching interior sphere is bounded by a multiple of the speed as long as the solution exists. The proof uses a maximum principle applied to a function of two points on the evolving hypersurface. We illustrate the techniques required for dealing with such functions in a proof of the known containment principle for flows of hypersurfaces.