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2014 Convexity estimates for hypersurfaces moving by convex curvature functions
Ben Andrews, Mathew Langford, James McCoy
Anal. PDE 7(2): 407-433 (2014). DOI: 10.2140/apde.2014.7.407


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.


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Ben Andrews. Mathew Langford. James McCoy. "Convexity estimates for hypersurfaces moving by convex curvature functions." Anal. PDE 7 (2) 407 - 433, 2014.


Received: 21 December 2012; Accepted: 23 July 2013; Published: 2014
First available in Project Euclid: 20 December 2017

zbMATH: 1294.53058
MathSciNet: MR3218814
Digital Object Identifier: 10.2140/apde.2014.7.407

Primary: 53C44
Secondary: 35K55

Keywords: convexity estimates , curvature flows , fully nonlinear

Rights: Copyright © 2014 Mathematical Sciences Publishers


Vol.7 • No. 2 • 2014
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