Trace heat kernel asymptotics in 3D contact sub-Riemannian geometry

Abstract: In this paper we study the small time asymptotics for the heat kernel on a sub-Riemannian manifold, using a perturbative approach. We then explicitly compute, in the case of a 3D contact structure, the first two coefficients of the small time asymptotics expansion of the heat kernel on the diagonal, expressing them in terms of the two basic functional invariants $\chi$ and $\kappa$ defined on a 3D contact structure.