Convergence of the volume preserving fractional mean curvature flow for convex sets

We prove that the volume preserving fractional mean curvature flow startingfrom a convex set does not develop singularities along the flow. By the recent result ofCesaroni-Novaga [6] this then implies that the flow converges to a ball exponentially fast.In the proof we show that the a priori estimates due to Cinti-Sinestrari-Valdinoci [10] implytheC1+α-regularity of the flow and then provide a regularity argument which improves thistoC2+α-regularity of the flow. The regularity step fromC1+αintoC2+αdoes not rely onconvexity and can be adopted to more general setting