Noncollapsing for Curvature Flows with Inhomogeneous Speeds

Abstract

We study closed, embedded hypersurfaces in Euclidean space evolving by fully nonlinear curvature flows, whose speed is given by a symmetric, monotone increasing, $1$-homogeneous, positive underlying speed function $F$ composed with a modulating function $\Psi$. Under the assumption that $F$ is convex or inverse-concave and that $\Psi$ satisfies the corresponding structural conditions, we establish exterior noncollapsing estimates for the flow. The main difficulty stems from the nonlinearity of the evolution equation satisfied in the viscosity sense by the exscribed curvature, whereas in previous works it is a solution to the linearized flow. Moreover, in the case where $F$ is inverse-concave, we refine Andrews and Langford's argument for the interior case.