Heat kernel asymptotics on sub-Riemannian manifolds with symmetries and applications to the bi-Heisenberg group

Abstract

By adapting a technique of Molchanov, we obtain the heat kernel asymptotics at the sub-Riemannian cut locus, when the cut points are reached by an $r$-dimensional parametric family of optimal geodesics. We apply these results to the bi-Heisenberg group, that is, a nilpotent left-invariant sub-Rieman\-nian structure on ${\mathbb{R}}^5$ depending on two real parameters ${\alpha }_1$ and ${\alpha }_2$. We develop some results about its geodesics and heat kernel associated to its sub-Laplacian and we illuminate some interesting geometric and analytic features appearing when one compares the isotropic (${\alpha }_1={\alpha }_2$) and the non-isotropic cases (${\alpha }_1\ne {\alpha }_2$). In particular, we give the exact structure of the cut locus, and we get the complete small-time asymptotics for its heat kernel.

17 pages, 1 figure