期刊
ANALYSIS & PDE
卷 7, 期 2, 页码 407-433出版社
MATHEMATICAL SCIENCE PUBL
DOI: 10.2140/apde.2014.7.407
关键词
convexity estimates; curvature flows; fully nonlinear
资金
- ARC Discovery Projects [DP0556211, DP120100097]
- Australian Research Council [DP0556211] Funding Source: Australian Research Council
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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