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

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 m × Snm. This result complements the convexity estimate we proved with McCoy [Anal. PDE 7:2 (2014), 407–433] for the same class of flows.

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