Analysis & PDE
- Anal. PDE
- Volume 7, Number 2 (2014), 407-433.
Convexity estimates for hypersurfaces moving by convex curvature functions
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.
Anal. PDE, Volume 7, Number 2 (2014), 407-433.
Received: 21 December 2012
Accepted: 23 July 2013
First available in Project Euclid: 20 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
Secondary: 35K55: Nonlinear parabolic equations
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