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

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 \( C^{1,\alpha }\) $C^{1,\alpha }$ 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 manifolds.