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

En adaptant une technique de Molchanov, nous obtenons le développement en temps petit du noyau de la chaleur au lieu de coupure sous-riemannien, quand les points de coupure sont rejoints par une famille à $r$ paramètres de géodésiques optimales. Nous appliquons ces résultats au cas du groupe de bi-Heisenberg, un exemple de structure sous-riemannienne nilpotente, invariante à gauche sur ${ℝ}^5$ qui dépend de deux paramètres réels ${\alpha }_1$ et ${\alpha }_2$. Nous décrivons des résultats concernants ses géodésiques et le noyau de la chaleur associé au sous-laplacien et nous mettons en évidence des propriétés géométriques et analytiques qui apparaissent quand on compare le cas isotrope (${\alpha }_1={\alpha }_2$) au cas non isotrope (${\alpha }_1\ne {\alpha }_2$). Notamment, nous obtenons la structure exacte du lieu de coupure avec la description complète du développement en temps petit du noyau de la chaleur.

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-Riemannian structure on ${ℝ}^5$ 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.

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