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 nonlinear curvature flows, generalising the cylindrical estimate of Huisken and Sinestrari [Invent. Math. 175:1 (2009), 1-14, §5] for the mean curvature flow. More precisely, we show, for the class of flows considered, that, at points where the curvature is becoming large, an (m+1)-convex (0 ≤ m ≤ n - 2) solution either becomes strictly m-convex or its Weingarten map becomes that of a cylinder Rm × Sn-m. This result complements the convexity estimate we proved with McCoy [Anal. PDE 7:2 (2014), 407-433] for the same class of flows.