Publication Details

Andrews, B., Langford, M. & McCoy, J. (2014). Convexity estimates for hypersurfaces moving by convex curvature functions. Analysis and PDE, 7 (2), 407-433.


We consider the evolution of compact hypersurfaces by fully non-linear, parabolic curvature ows for which the normal speed is given by a smooth, convex, degree one homoge- neous 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 ow. The result extends the convexity estimate [HS99b] of Huisken and Sinestrari for the mean curvature ow 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.

Grant Number

ARC/DP0556211, ARC/DP120100097