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, Serie 6, Volume 28 (2019) no. 4, pp. 707-732.

Abstract
Abstract

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.

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.

Received: 2016-06-02
Accepted: 2017-07-23
Published online: 2019-12-09

Zbl

DOI: 10.5802/afst.1613

Author's affiliations:

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)

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@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},
     pages = {707--732},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 28},
     number = {4},
     year = {2019},
     doi = {10.5802/afst.1613},
     zbl = {1270.53066},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1613/}
}

TY  - JOUR
AU  - Davide Barilari
AU  - Ugo Boscain
AU  - Robert W. Neel
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
SP  - 707
EP  - 732
VL  - 28
IS  - 4
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1613/
DO  - 10.5802/afst.1613
LA  - en
ID  - AFST_2019_6_28_4_707_0
ER  -

%0 Journal Article
%A Davide Barilari
%A Ugo Boscain
%A Robert W. Neel
%T Heat kernel asymptotics on sub-Riemannian manifolds with symmetries and applications to the bi-Heisenberg group
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2019
%P 707-732
%V 28
%N 4
%I Université Paul Sabatier, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1613/
%R 10.5802/afst.1613
%G en
%F AFST_2019_6_28_4_707_0

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, Serie 6, Volume 28 (2019) no. 4, pp. 707-732. doi : 10.5802/afst.1613. https://afst.centre-mersenne.org/articles/10.5802/afst.1613/

References
Cited by

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

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

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

[4] Andrei Agrachev; Davide Barilari; Ugo Boscain 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

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

[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., Volume 256 (2009) no. 8, pp. 2621-2655 | DOI | MR | Zbl

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

[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., Volume 15 (2017), pp. 4639-4672 | Zbl

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

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

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

[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., Volume 121 (1996) no. 2, pp. 288-345 | DOI | MR | Zbl

[13] Gérard Ben Arous 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 | DOI | Numdam | MR | Zbl

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

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

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

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

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

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

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

Cited by Sources: