Neckpinch singularities in fractional mean curvature flows

TL;DR

This paper demonstrates that in higher dimensions, fractional mean curvature flows can develop singularities before shrinking to a point, contrasting classical behavior and providing new insights into nonlocal geometric flows.

Contribution

It constructs explicit examples of singularity formation in fractional mean curvature flows for dimensions greater than two, extending classical results and offering counterexamples in the planar case.

Findings

Existence of singularities in fractional mean curvature flows for n > 2

Generalization of Grayson's classical results to higher dimensions

Counterexample to Grayson's theorem in the nonlocal setting for n=2

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.