Mean curvature flow of star-shaped hypersurfaces

2015/08/05 by Longzhi Lin, Lin, Longzhi
Mathematics · #35K55 #53C44 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1508.01225

openalex publication_date 2015/08/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 R³. We prove in this paper that this is true for the mean curvature flow of star-shaped hypersurfaces in Rⁿ⁺¹ in arbitrary dimension n≥ 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].

Citations

• Uniqueness of blowups and Lojasiewicz inequalities

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