Heat kernel asymptotics on sub-Riemannian manifolds with symmetries and applications to the bi-Heisenberg group
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 28 (2019) no. 4, pp. 707-732.

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 5 qui dépend de deux paramètres réels α 1 et α 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 (α 1 =α 2 ) au cas non isotrope (α 1 α 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 5 depending on two real parameters α 1 and α 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 (α 1 =α 2 ) and the non-isotropic cases (α 1 α 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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DOI : 10.5802/afst.1613
Barilari, Davide 1 ; Boscain, Ugo 2 ; Neel, Robert W. 3

1 IMJ-PRG, Université Paris Diderot, UMR CNRS 7586, Paris (France)
2 CNRS, Sorbonne Universités, Laboratoire LJLL, Inria de Paris team CAGE, Paris (France)
3 Department of Mathematics, Lehigh University, Bethlehem, PA (USA)
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Barilari, Davide; Boscain, Ugo; Neel, Robert W. Heat kernel asymptotics on sub-Riemannian manifolds with symmetries and applications to the bi-Heisenberg group. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 28 (2019) no. 4, pp. 707-732. doi : 10.5802/afst.1613. http://www.numdam.org/articles/10.5802/afst.1613/

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