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.

Cinti, E., Sinestrari, C., Valdinoci, E. (2018). Neckpinch singularities in fractional mean curvature flows. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 146(6), 2637-2646 [10.1090/proc/14002].

Neckpinch singularities in fractional mean curvature flows

Sinestrari, Carlo;Valdinoci, Enrico
2018-01-01

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.
2018
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05 - ANALISI MATEMATICA
Settore MAT/03 - GEOMETRIA
English
Con Impact Factor ISI
Fractional mean curvature flow; Fractional perimeter
http://www.ams.org/journals/proc/2018-146-06/S0002-9939-2018-14002-2/S0002-9939-2018-14002-2.pdf
Cinti, E., Sinestrari, C., Valdinoci, E. (2018). Neckpinch singularities in fractional mean curvature flows. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 146(6), 2637-2646 [10.1090/proc/14002].
Cinti, E; Sinestrari, C; Valdinoci, E
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/197611
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