Asymptotiques en temps petit du noyau de la chaleur des métriques riemanniennes et sous-riemanniennes

Barilari, Davide¹  ; Boscain, Ugo¹  ; Charlot, Grégoire²  ; Neel, Robert W.³

¹ CNRS, CMAP Ecole Polytechnique and equipe INRIA GECO Saclay-Île-de-France, Paris, France
² Institut Fourier, UMR 5582, Université Grenoble 1 and equipe INRIA GECO Saclay-Île-de-France, Paris, France
³ Department of Mathematics, Lehigh University, Bethlehem, PA, USA

Séminaire de théorie spectrale et géométrie, Tome 31 (2012-2014), pp. 55-70.

Résumé (VO)

Nous établissons l’asymptotique en temps petit du noyau de la chaleur au lieu de coupure dans les situations génériques, en géométrie riemannienne en dimension inférieure ou égale à 5, en géométrie sous-riemannienne de contact en dimension 3 ou de quasi-contact en dimension 4. La preuve nous permet de montrer qu’en dimension inférieure ou égale à 5 les seules singularités d’une application exponentielle riemannienne générique qui peuvent apparaître le long d’une géodésique minimisante sont $A_3$ et $A_5$.

Abstract

Abstract. We provide the small-time asymptotics of the heat kernel at the cut locus in three cases: generic Riemannian manifolds in dimension less or equal to 5, generic 3D contact and 4D quasi-contact sub-Riemannian manifolds (close to the starting point). As a byproduct we show that, for generic Riemannian manifolds of dimension less or equal to 5, the only possible singularities of the exponential map along a minimizing geodesic are $A_3$ and $A_5$.

Détail

DOI : 10.5802/tsg.294

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