WIAS Preprint No. 2282, (2016)

Neckpinch singularities in fractional mean curvature flows



Authors

  • Cinti, Eleonora
  • Sinestrari, Carlo
  • Valdinoci, Enrico
    ORCID: 0000-0001-6222-2272

2010 Mathematics Subject Classification

  • 53C44 35R11

Keywords

  • fractional perimeter, fractional mean curvature flow

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 [17], 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.

Appeared in

  • Proc. Amer. Math. Soc., 146 (2018) pp. 2637--2646.

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