#### Vol. 7, No. 5, 2014

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Cylindrical estimates for hypersurfaces moving by convex curvature functions

### Ben Andrews and Mat Langford

Vol. 7 (2014), No. 5, 1091–1107
##### 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 ${ℝ}^{m}×{S}^{n-m}$. This result complements the convexity estimate we proved with McCoy [Anal. PDE 7:2 (2014), 407–433] for the same class of flows.

##### Keywords
curvature flows, cylindrical estimates, fully nonlinear, convexity estimates
##### Mathematical Subject Classification 2010
Primary: 53C44, 35K55, 58J35
##### Milestones
Received: 8 October 2013
Accepted: 30 June 2014
Published: 27 September 2014
##### Authors
 Ben Andrews Mathematical Sciences Institute Australian National University ACT 0200 Australia Mathematical Sciences Center Tsinghua University Beijing 100084 China Mat Langford Mathematical Sciences Institute Australian National University ACT 0200 Australia Fachbereich Mathematik und Statistik Universität Konstanz 78457 Konstanz Germany