Neckpinch singularities in fractional mean curvature flows

Eleonora Cinti, Carlo Sinestrari, Enrico Valdinoci

Research output: Contribution to journalArticlepeer-review

18 Citations (Scopus)


In this paper we consider the evolution of boundaries of sets by a fractional mean curvature flow. We show that for any dimension n ≥ 2, there exist embedded hypersurfaces in Rn which develop a singularity without shrinking to a point. Such examples are well known for the classical mean curvature flow for n ≥ 3. Interestingly, when n = 2, our result provides instead a counterexample in the nonlocal framework to the well-known Grayson’s Theorem, which states that any smooth embedded curve in the plane evolving by (classical) MCF shrinks to a point. The essential step in our construction is an estimate which ensures that a suitably small perturbation of a thin strip has positive fractional curvature at every boundary point.

Original languageEnglish
Pages (from-to)2637-2646
Number of pages10
JournalProceedings of the American Mathematical Society
Issue number6
Publication statusPublished - 2018
Externally publishedYes


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