Cylindrical estimates for hypersurfaces moving by convex curvature functions

Abstract

We prove a complete family of `cylindrical estimates' for solutions of a class of fully non-linear curvature flows, generalising the cylindrical estimate of Huisken-Sinestrari [HS09, Section 5] for the mean curvature flow. More precisely, we show that, for the class of flows considered, an (m + 1)-convex (0 ≤ m ≤ n - 2) solution becomes either strictly m-convex, or its Weingarten map approaches that of a cylinder Rᵐ x Sⁿ⁻ ᵐ at points where the curvature is becoming large. This result complements the convexity estimate proved in [ALM13] for the same class of flows. Show more