Curvature measures and the sub-Riemannian Gauss-Bonnet theorem

We adopt a measure-theoretic perspective on the Riemannian approximation scheme proving a sub-Riemannian Gauss-Bonnet theorem for surfaces in 3D contact manifolds. We show that the zero-order term in the limit is a singular measure supported on isolated characteristic points. In particular, this provides a unified interpretation of previous results. Moreover we give natural geometric conditions under which our result holds, namely when the surface admits characteristic points of finite order of degeneracy. This notion, which we introduce, extends the concept of mildly degenerate characteristic points for the Heisenberg group. As a byproduct, we prove that the mean curvature around an isolated characteristic point of finite order of degeneracy is locally integrable. In particular, this positively answers a question for analytic surfaces in every analytic 3D contact manifold.