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 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 qui dépend de deux paramètres réels et . 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 () au cas non isotrope (). 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 -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 depending on two real parameters and . 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 () and the non-isotropic cases (). In particular, we give the exact structure of the cut locus, and we get the complete small-time asymptotics for its heat kernel.
@article{AFST_2019_6_28_4_707_0, author = {Barilari, Davide and Boscain, Ugo and Neel, Robert W.}, 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}, number = {4}, year = {2019}, doi = {10.5802/afst.1613}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/afst.1613/} }
TY - JOUR AU - Barilari, Davide AU - Boscain, Ugo AU - Neel, Robert W. TI - Heat kernel asymptotics on sub-Riemannian manifolds with symmetries and applications to the bi-Heisenberg group JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2019 DA - 2019/// SP - 707 EP - 732 VL - Ser. 6, 28 IS - 4 PB - Université Paul Sabatier, Toulouse UR - http://archive.numdam.org/articles/10.5802/afst.1613/ UR - https://doi.org/10.5802/afst.1613 DO - 10.5802/afst.1613 LA - en ID - AFST_2019_6_28_4_707_0 ER -
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://archive.numdam.org/articles/10.5802/afst.1613/
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