Analysis & PDE

  • Anal. PDE
  • Volume 7, Number 2 (2014), 407-433.

Convexity estimates for hypersurfaces moving by convex curvature functions

Ben Andrews, Mathew Langford, and James McCoy

Full-text: Open access

Abstract

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.

Article information

Source
Anal. PDE, Volume 7, Number 2 (2014), 407-433.

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

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731494

Digital Object Identifier
doi:10.2140/apde.2014.7.407

Mathematical Reviews number (MathSciNet)
MR3218814

Zentralblatt MATH identifier
1294.53058

Subjects
Primary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
Secondary: 35K55: Nonlinear parabolic equations

Keywords
convexity estimates curvature flows fully nonlinear

Citation

Andrews, Ben; Langford, Mathew; McCoy, James. Convexity estimates for hypersurfaces moving by convex curvature functions. Anal. PDE 7 (2014), no. 2, 407--433. doi:10.2140/apde.2014.7.407. https://projecteuclid.org/euclid.apde/1513731494


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