Davide Barilari (Università degli Studi di Padova) – Curvature measures and the Sub-Riemannian Gauss-Bonnet Theorem

Sub-Riemannian geometries can be seen as limits of Riemannian metrics. In this talk, we focus on the behaviour of the Gauss-Bonnet theorem under this Riemannian approximation scheme for surfaces embedded in three-dimensional spaces. Adopting a measure-theoretic viewpoint, we prove that the limit Gaussian curvature measure of such a surface is singular and supported on its isolated characteristic points. We identify natural geometric conditions under which this behavior occurs, namely when the surface admits characteristic points of finite order of degeneracy.