Neckpinch singularities in fractional mean curvature flows
- Cinti, Eleonora
- Sinestrari, Carlo
- Valdinoci, Enrico
2010 Mathematics Subject Classification
- 53C44 35R11
- fractional perimeter, fractional mean curvature flow
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.
- Proc. Amer. Math. Soc., 146 (2018) pp. 2637--2646.