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Convexity estimates for
hypersurfaces moving by convex curvature functions
Ben Andrews, Mat Langford and James McCoy |
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Vol. 7 (2014), No. 2, 407–433
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Abstract |
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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. |
Keywords
convexity estimates, curvature flows, fully nonlinear
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Mathematical Subject Classification 2010
Primary: 53C44
Secondary: 35K55
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Milestones
Received: 21 December 2012
Accepted: 23 July 2013
Published: 30 May 2014
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Authors |
| Ben Andrews | |
| Mathematical Sciences
Institute Australian National University Building 27 Acton ACT 0200 Australia |
Mathematical Sciences Center Tsinghua University Beijing 100084 China |
| Mat Langford | |
| Mathematical Sciences
Institute Australian National University Building 27 Acton ACT 0200 Australia |
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| James McCoy | |
| Institute for Mathematics and its
Applications School of Mathematics and Applied Statistics University of Wollongong Wollongong NSW 2522 Australia |
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