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

Davide Barilari; Ugo Boscain; Robert W. Neel

doi:10.5802/afst.1613

Davide Barilari ¹ ; Ugo Boscain ² ; Robert W. Neel 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)

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 28 (2019) no. 4, pp. 707-732.

Résumé
Abstract

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 $α_{1}$ 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 ( $α_{1} \neq α_{2}$ ). 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 $ℝ^{5}$ 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 ( $α_{1} \neq α_{2}$ ). In particular, we give the exact structure of the cut locus, and we get the complete small-time asymptotics for its heat kernel.

Reçu le : 2016-06-02
Accepté le : 2017-07-23
Publié le : 2019-12-09

Zbl

DOI : 10.5802/afst.1613

Affiliations des auteurs :

Davide Barilari ¹ ; Ugo Boscain ² ; Robert W. Neel 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)

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Heat kernel asymptotics on {sub-Riemannian} manifolds with symmetries and applications to the {bi-Heisenberg} group},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {707--732},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 28},
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Davide Barilari; Ugo Boscain; Robert W. Neel. 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. https://afst.centre-mersenne.org/articles/10.5802/afst.1613/

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