Vol. 7, No. 2, 2014

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Convexity estimates for hypersurfaces moving by convex curvature functions

Ben Andrews, Mat Langford and James McCoy

Vol. 7 (2014), No. 2, 407–433

We consider the evolution of compact hypersurfaces by fully nonlinear, parabolic curvature flows for which the normal speed is given by a smooth, convex, degree-one homogeneous function of the principal curvatures. We prove that solution hypersurfaces on which the speed is initially positive become weakly convex at a singularity of the flow. The result extends the convexity estimate of Huisken and Sinestrari [Acta Math. 183:1 (1999), 45–70] for the mean curvature flow to a large class of speeds, and leads to an analogous description of “type-II” singularities. We remark that many of the speeds considered are positive on larger cones than the positive mean half-space, so that the result in those cases also applies to non-mean-convex initial data.

convexity estimates, curvature flows, fully nonlinear
Mathematical Subject Classification 2010
Primary: 53C44
Secondary: 35K55
Received: 21 December 2012
Accepted: 23 July 2013
Published: 30 May 2014
Ben Andrews
Mathematical Sciences Institute
Australian National University
Building 27
Acton ACT 0200
Mathematical Sciences Center
Tsinghua University
Beijing 100084
Mat Langford
Mathematical Sciences Institute
Australian National University
Building 27
Acton ACT 0200
James McCoy
Institute for Mathematics and its Applications
School of Mathematics and Applied Statistics
University of Wollongong
Wollongong NSW 2522