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arXiv · 2307.03912

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

Vesa Julin·Domenico Angelo La Manna
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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.

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Vesa Julin, Domenico Angelo La Manna. 2023-07-08. Convergence of the volume preserving fractional mean curvature flow for convex sets. https://arxiv.org/abs/2307.03912

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