SMALL-TIME HEAT KERNEL ASYMPTOTICS AT THE SUB-RIEMANNIAN CUT LOCUS

For a sub-Riemannian manifold provided with a smooth volume, we relate the small-time asymptotics of the heat kernel at a point y of the cut locus from x with roughly "how much" y is conjugate to x. This is done under the hypothesis that all minimizers connecting x to y are strongly normal, i.e. all pieces of the trajectory are not abnormal. Our result is a refinement of the one of Leandre 4t log p(t)(x, y) -> -d(2)(x, y) for t -> 0, in which only the leading exponential term is detected. Our results are obtained by extending an idea of Molchanov from the Riemannian to the sub-Riemannian case, and some details we get appear to be new even in the Riemannian context. These results permit us to obtain properties of the sub-Riemannian distance starting from those of the heat kernel and vice versa. For the Grushin plane endowed with the Euclidean volume, we get the expansion p(t) (x, y) similar to t(-5/4) exp(-d(2)(x, y)/4t) where y is reached from a - Riemannian point x by a minimizing geodesic which is conjugate at y.