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Neckpinch singularities in fractional mean curvature flows

Authors: Eleonora Cinti, Carlo Sinestrari and Enrico Valdinoci
Journal: Proc. Amer. Math. Soc. 146 (2018), 2637-2646
MSC (2010): Primary 53C44, 35R11
Published electronically: February 21, 2018
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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\geqslant 2$, there exist embedded hypersurfaces in $ \mathbb{R}^n$ which develop a singularity without shrinking to a point. Such examples are well known for the classical mean curvature flow for $ n \geqslant 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.

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Additional Information

Eleonora Cinti
Affiliation: Weierstraß Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin, Germany
Address at time of publication: Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy

Carlo Sinestrari
Affiliation: Dipartimento di Ingegneria Civile e Ingegneria Informatica, Università di Roma Tor Vergata, Via del Politecnico, 00133 Rome, Italy

Enrico Valdinoci
Affiliation: School of Mathematics and Statistics, University of Melbourne, 813 Swanston Street, Parkville VIC 3010, Australia – and – Dipartimento di Matematica, Università di Milano, Via Cesare Saldini 50, 20133 Milan, Italy – and – Weierstraß Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin, Germany

Keywords: Fractional perimeter, fractional mean curvature flow.
Received by editor(s): July 5, 2017
Published electronically: February 21, 2018
Additional Notes: The first author was supported by grants MTM2011-27739-C04-01 (Spain), 2009SGR345 (Catalunya), and by the ERC Starting Grant “AnOptSetCon” n. 258685.
The first and third authors were supported by the ERC Starting Grant “EPSILON”, Elliptic PDE’s and Symmetry of Interfaces and Layers for Odd Nonlinearities n. 277749.
The second author was supported by the group GNAMPA of INdAM Istituto Nazionale di Alta Matematica.
Communicated by: Lei Ni
Article copyright: © Copyright 2018 American Mathematical Society

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