Mean curvature flow of star-shaped hypersurfaces

In the last 15 years, the series of works of White and Huisken–Sinestrari yield that the blowup limits at singularities are convex for the mean curvature flow of mean convex hypersurfaces. In 1998 Smoczyk [20] showed that, among others, the blowup limits at singularities are convex for the mean curvature flow starting from a closed star-shaped surface in $\mathbf{R}^3$.We prove in this paper that this is still true for the mean curvature flow of star-shaped hypersurfaces in $\mathbf{R}^{n+1}$ in arbitrary dimension $n\geq 2$. In fact, this holds for a much more general class of initial hypersurfaces. In particular, this implies that the mean curvature flow of star-shaped hypersurfaces is generic in the sense of Colding–Minicozzi [6].

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The author was partially supported by a Faculty Research Grant awarded by the Committee on Research from the University of California at Santa Cruz.

Received 2 February 2016

Accepted 11 March 2018

Published 2 December 2020