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arXiv · 1607.08032

Neckpinch singularities in fractional mean curvature flows

Eleonora Cinti·Carlo Sinestrari·Enrico Valdinoci
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Abstract

In this paper we consider the evolution of sets by a fractional mean curvature flow. Our main result states that for any dimension $n > 2$, there exists an embedded surface in $\mathbb R^n$ evolving by fractional mean curvature flow, which developes a singularity before it can shrink to a point. When $n > 3$ this result generalizes the analogue result of Grayson for the classical mean curvature flow. Interestingly, when $n = 2$, our result provides instead a counterexample in the nonlocal framework to the well known Grayson Theorem, which states that any smooth embedded curve in the plane evolving by (classical) MCF shrinks to a point.

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Eleonora Cinti, Carlo Sinestrari, Enrico Valdinoci. 2016-07-27. Neckpinch singularities in fractional mean curvature flows. https://arxiv.org/abs/1607.08032

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