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.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/afst.1613
@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/

[1] Agrachev, Andrei Exponential mappings for contact sub-Riemannian structures, J. Dyn. Control Syst., Volume 2 (1996) no. 3, pp. 321-358 | Article | MR 1403262 | Zbl 0941.53022

[2] Agrachev, Andrei Compactness for sub-Riemannian length-minimizers and subanalyticity, Rend. Semin. Mat., Torino, Volume 56 (1998) no. 4, pp. 1-12 | MR 1845741 | Zbl 1039.53038

[3] Agrachev, Andrei; Barilari, Davide; Boscain, Ugo On the Hausdorff volume in sub-Riemannian geometry, Calc. Var. Partial Differ. Equ., Volume 43 (2012) no. 3-4, pp. 355-388 | Article | MR 2875644

[4] Agrachev, Andrei; Barilari, Davide; Boscain, Ugo Introduction to geodesics in sub-Riemannian geometry, Geometry, analysis and dynamics on sub-Riemannian manifolds. Vol. II (EMS Series of Lectures in Mathematics), European Mathematical Society, 2016, pp. 1-83 | Zbl 1362.53001

[5] Agrachev, Andrei; Barilari, Davide; Boscain, Ugo A Comprehensive Introduction to sub-Riemannian Geometry (2019) (Cambridge University Press, in press) | Zbl 07073879

[6] Agrachev, Andrei; Boscain, Ugo; Gauthier, Jean-Paul; Rossi, Francesco The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups, J. Funct. Anal., Volume 256 (2009) no. 8, pp. 2621-2655 | Article | MR 2502528 | Zbl 1165.58012

[7] Agrachev, Andrei; Sachkov, Yuri L. Control theory from the geometric viewpoint, Encyclopaedia of Mathematical Sciences, 87, Springer, 2004 (Control Theory and Optimization, II) | MR 2062547 | Zbl 1062.93001

[8] Barilari, Davide; Boscain, Ugo; Charlot, Grégoire; Neel, Robert W. On the heat diffusion for generic Riemannian and sub-Riemannian structures, Int. Math. Res. Not., Volume 15 (2017), pp. 4639-4672 | Zbl 1405.58006

[9] Barilari, Davide; Boscain, Ugo; Gauthier, Jean-Paul On 2-step, corank 2, nilpotent sub-Riemannian metrics, SIAM J. Control Optimization, Volume 50 (2012) no. 1, pp. 559-582 | Article | MR 2888278 | Zbl 1243.53064

[10] Barilari, Davide; Boscain, Ugo; Neel, Robert W. Small-time heat kernel asymptotics at the sub-Riemannian cut locus, J. Differ. Geom., Volume 92 (2012) no. 3, pp. 373-416 | Article | MR 3005058 | Zbl 1270.53066

[11] Barilari, Davide; Rizzi, Luca A formula for Popp’s volume in sub-Riemannian geometry, Anal. Geom. Metr. Spaces, Volume 1 (2013), pp. 42-57 | Article | MR 3108867 | Zbl 1260.53062

[12] Beals, Richard; Gaveau, Bernard; Greiner, Peter The Green function of model step two hypoelliptic operators and the analysis of certain tangential Cauchy Riemann complexes, Adv. Math., Volume 121 (1996) no. 2, pp. 288-345 | Article | MR 1402729 | Zbl 0858.43009

[13] Ben Arous, Gérard Développement asymptotique du noyau de la chaleur hypoelliptique hors du cut-locus, Ann. Sci. Éc. Norm. Supér., Volume 21 (1988) no. 3, pp. 307-331 | Article | Numdam | MR 974408 | Zbl 0699.35047

[14] Ben Arous, Gérard Développement asymptotique du noyau de la chaleur hypoelliptique sur la diagonale, Ann. Inst. Fourier, Volume 39 (1989) no. 1, pp. 73-99 | Numdam | Zbl 0659.35024

[15] Bonfiglioli, Andrea; Lanconelli, Ermanno; Uguzzoni, Francesco Stratified Lie groups and potential theory for their sub-Laplacians, Springer Monographs in Mathematics, Springer, 2007 | Zbl 1128.43001

[16] Burago, Dmitri; Burago, Yuri; Ivanov, Sergei A course in metric geometry, Graduate Studies in Mathematics, 33, American Mathematical Society, 2001 | MR 1835418 | Zbl 1232.53037

[17] Estrada, Ricardo; Kanwal, Ram P. A distributional approach to asymptotics. Theory and applications, Birkhäuser Advanced Texts. Basler Lehrbücher, Birkhäuser, 2002 | Zbl 1033.46031

[18] Inahama, Yuzuru; Taniguchi, Setsuo Short time full asymptotic expansion of hypoelliptic heat kernel at the cut locus, Forum Math. Sigma, Volume 5 (2017), e16, 74 pages | MR 3669328 | Zbl 1369.60040

[19] Molčanov, Stanislav A. Diffusion processes, and Riemannian geometry, Usp. Mat. Nauk, Volume 30 (1975) no. 1, pp. 3-59 | MR 413289

[20] Strichartz, Robert S. Sub-Riemannian geometry, J. Differ. Geom., Volume 24 (1986) no. 2, pp. 221-263 | Article | MR 862049 | Zbl 0609.53021

Cité par Sources :