Calculus of Variations and Geometric Measure Theory

D. Barilari - U. Boscain - R. W. Neel

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

created by barilari on 03 Jun 2016
modified on 16 Apr 2021


Published Paper

Inserted: 3 jun 2016
Last Updated: 16 apr 2021

Journal: Annales de la Faculté des Sciences de Toulouse
Volume: 28
Number: 4
Pages: 707-732
Year: 2019


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.