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

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 note is the introduction of a concept of 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-Nhieu in [DGN12], proving that the mean curvature of a real-analytic surface with discrete characteristic set is locally integrable.