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

TL;DR

This paper derives heat kernel asymptotics on sub-Riemannian manifolds with symmetries, focusing on the bi-Heisenberg group, and explores geometric and analytic features distinguishing isotropic and non-isotropic cases.

Contribution

It adapts Molchanov's technique to obtain heat kernel asymptotics at the cut locus for manifolds with symmetries, specifically applying to the bi-Heisenberg group.

Findings

Exact structure of the cut locus for the bi-Heisenberg group

Complete small-time heat kernel asymptotics

Differences between isotropic and non-isotropic cases

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