Neckpinch Singularities In Fractional Mean Curvature Flows
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Neckpinch Singularities in Fractional Mean Curvature Flows
Author | : Eleonora Cinti |
Publisher | : |
Total Pages | : |
Release | : 2016 |
Genre | : |
ISBN | : |
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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.
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