Heat kernel asymptotics on subRiemannian manifolds with symmetries and applications to the biHeisenberg group
Abstract
By adapting a technique of Molchanov, we obtain the heat kernel asymptotics at the subRiemannian cut locus, when the cut points are reached by an $r$dimensional parametric family of optimal geodesics. We apply these results to the biHeisenberg group, that is, a nilpotent leftinvariant subRieman\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 subLaplacian and we illuminate some interesting geometric and analytic features appearing when one compares the isotropic ($\alpha_{1}=\alpha_{2}$) and the nonisotropic cases ($\alpha_{1}\neq \alpha_{2}$). In particular, we give the exact structure of the cut locus, and we get the complete smalltime asymptotics for its heat kernel.
 Publication:

arXiv eprints
 Pub Date:
 June 2016
 DOI:
 10.48550/arXiv.1606.01159
 arXiv:
 arXiv:1606.01159
 Bibcode:
 2016arXiv160601159B
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Differential Geometry;
 Mathematics  Optimization and Control
 EPrint:
 17 pages, 1 figure