Off diagonal asymptotics of hypoelliptic diffusion equations and singular Riemannian geometry

The major purpose of this paper is to investigate some small time asymptotic properties of certain diffusion equations of the form where Δ is in a class of second order hypoelliptic differential operators on a connected m-dimensional manifold M, and where Γ_x(y) is the Dirac Γ-function in the variable y ∈ M supported at the point x ∈ M.