$C^{1,\alpha}$ regularity of hypersurfaces of bounded nonlocal mean curvature in Riemannian manifolds

Abstract

Let $(M,g)$ be a smooth connected Riemannian manifold. We show an improvement of flatness theorem for hypersurfaces of $M$ of bounded nonlocal mean curvature in the viscosity sense. It implies local $ C^{1,\alpha}$ regularity of these hypersurfaces provided that they are sufficiently flat. It extends a result of Caffarelli, Roquejoffre and Savin in the Euclidean setting to the case of arbitrary Riemannian manifolds.