Integrability of the sub-Riemannian mean curvature at degenerate characteristic points in the Heisenberg group

Abstract

Abstract We address the problem of integrability of the sub-Riemannian mean curvature of an embedded hypersurface around isolated characteristic points. The main contribution of this paper is the introduction of a concept of a mildly degenerate characteristic point for a smooth surface of the Heisenberg group, in a neighborhood of which the sub-Riemannian mean curvature is integrable (with respect to the perimeter measure induced by the Euclidean structure). As a consequence, we partially answer to a question posed by Danielli, Garofalo and Nhieu in [D. Danielli, N. Garofalo and D. M. Nhieu, Integrability of the sub-Riemannian mean curvature of surfaces in the Heisenberg group, Proc. Amer. Math. Soc. 140 2012, 3, 811–821], proving that the mean curvature of a real-analytic surface with discrete characteristic set is locally integrable.