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 $\left(m+1\right)$-convex ($0\le m\le n-2$) solution either becomes strictly $m$-convex or its Weingarten map becomes that of a cylinder ${ℝ}^m\times S^{n-m}$. This result complements the convexity estimate we proved with McCoy [Anal. PDE 7:2 (2014), 407–433] for the same class of flows.