Some recent results in the study of fractional mean curvature flow

We study a geometric flow driven by the fractional mean curvature (FMC). The notion of fractional mean curvature arises naturally when performing the first variation of the fractional perimeter functional (introduced by Caffarelli, Roquejoffre, and Savin). More precisely, we show the existence of surfaces which develope neckpinch singularities, in any dimension n ≥ 2. Interestingly, in dimension n=2 our result gives a counterexample to Greyson Theorem for the classical mean curvature flow. The result has been obtained in collaboration with C. Sinestrari and E. Valdinoci.