Neckpinch singularities in fractional mean curvature flows

Eleonora Cinti, Carlo Sinestrari, Enrico Valdinoci

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

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
Volume146
Issue number6
DOIs
Publication statusPublished - 2018
Externally publishedYes

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Mean Curvature Flow
Fractional
Singularity
Shrinking
Small Perturbations
Hypersurface
Strip
Counterexample
Curvature
Curve
Theorem
Estimate

Cite this

Cinti, Eleonora ; Sinestrari, Carlo ; Valdinoci, Enrico. / Neckpinch singularities in fractional mean curvature flows. In: Proceedings of the American Mathematical Society. 2018 ; Vol. 146, No. 6. pp. 2637-2646.
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Neckpinch singularities in fractional mean curvature flows. / Cinti, Eleonora; Sinestrari, Carlo; Valdinoci, Enrico.

In: Proceedings of the American Mathematical Society, Vol. 146, No. 6, 2018, p. 2637-2646.

Research output: Contribution to journalArticle

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