Mean curvature flow of star-shaped hypersurfaces

Abstract

In 1998 Smoczyk [Smo98] 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 true for the mean curvature flow of star-shaped hypersurfaces in ${\mathbf{R}}^{n+1}$ in arbitrary dimension $n\ge 2$. 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 [CM12].

19 pages. arXiv admin note: substantial text overlap with arXiv:1304.0926 by other authors

References in corpus (1)

• Uniqueness of blowups and Lojasiewicz inequalities

Cited by in corpus (2)

• On mean curvature flow with forcing
• Mean curvature flow with generic initial data