Non-collapsing in fully non-linear curvature flows

Abstract

We consider compact, embedded hypersurfaces of Euclidean spaces evolving by fully non-linear 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 ball 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 balls. In particular, if the hypersurface moves with positive speed and the speed is concave in the principal curvatures, the curvature of the largest touching interior ball 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.