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

TL;DR

This paper proves that convex sets evolving under volume-preserving fractional mean curvature flow remain smooth and converge exponentially to a sphere, extending regularity results and avoiding singularities.

Contribution

It establishes $C^{2+eta}$ regularity for the flow without convexity assumptions and confirms exponential convergence to a sphere for convex sets.

Findings

Flow remains smooth without developing singularities.

Convex sets converge exponentially to a spherical shape.

Regularity improved from $C^{1+eta}$ to $C^{2+eta}$ without convexity.

Abstract

We prove that the volume preserving fractional mean curvature flow starting from a convex set does not develop singularities along the flow. By the recent result of Cesaroni-Novaga \cite{CN} this then implies that the flow converges to a ball exponentially fast. In the proof we show that the apriori estimates due to Cinti-Sinestrari-Valdinoci \cite{CSV2} imply the $C^{1+\alpha}$-regularity of the flow and then provide a regularity argument which improves this into $C^{2+\alpha}$-regularity of the flow. The regularity step from $C^{1+\alpha}$ into $C^{2+\alpha}$ does not rely on convexity and can probably be adopted to more general setting.