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

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 ($\alpha_{1}\neq \alpha_{2}$). In particular, we give the exact structure of the cut locus, and we get the complete small-time asymptotics for its heat kernel.