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Heat kernel asymptotics on sub-Riemannian manifolds with symmetries and applications to the bi-Heisenberg group
Davide Barilari; Ugo Boscain; Robert W. Neel
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 28 (2019) no. 4, p. 707-732

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.

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.

Received : 2016-06-03
Accepted : 2017-07-24
Published online : 2019-12-09
DOI : https://doi.org/10.5802/afst.1613
@article{AFST_2019_6_28_4_707_0,
     author = {Davide Barilari and Ugo Boscain and Robert W. Neel},
     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},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 28},
     number = {4},
     year = {2019},
     pages = {707-732},
     doi = {10.5802/afst.1613},
     language = {en},
     url = {https://afst.centre-mersenne.org/item/AFST_2019_6_28_4_707_0}
}
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, Serie 6, Volume 28 (2019) no. 4, pp. 707-732. doi : 10.5802/afst.1613. https://afst.centre-mersenne.org/item/AFST_2019_6_28_4_707_0/

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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