The Laplace-Beltrami operator in almost-Riemannian Geometry
[L’opérateur de Laplace-Beltrami en Géométrie presque-Riemannienne]
Annales de l'Institut Fourier, Tome 63 (2013) no. 5, pp. 1739-1770.

On étudie l’opérateur de Laplace-Beltrami dans des structures Riemanniennes généralisées sur des surfaces orientables pour lesquelles un repère orthonormé est donné par une paire de champs de vecteurs pouvant devenir colinéaires.

Sous l’hypothèse que la structure est génératrice avec des crochets de Lie d’ordre 2, on prouve que l’opérateur de Laplace-Beltrami est essentiellement autoadjoint et a un spectre discret. Par conséquent, une particule quantique ne peut pas traverser l’ensemble singulier (i.e., là où les champs de vecteurs deviennent colinéaires) et la chaleur ne peut pas diffuser à travers la singularité. Le phénomène est intéressant puisque lorsqu’on s’approche de l’ensemble singulier, toutes les quantités Riemanniennes explosent, mais les géodésiques sont encore bien définies et peuvent traverser l’ensemble singulier sans singularité.

Ce phénomène apparaît aussi dans des structures sous-Riemanniennes qui ne sont pas équirégulières, i.e., dont le vecteur de croissance dépend du point. On montre ce fait en analysant le cas Martinet.

We study the Laplace-Beltrami operator of generalized Riemannian structures on orientable surfaces for which a local orthonormal frame is given by a pair of vector fields that can become collinear.

Under the assumption that the structure is 2-step Lie bracket generating, we prove that the Laplace-Beltrami operator is essentially self-adjoint and has discrete spectrum. As a consequence, a quantum particle cannot cross the singular set (i.e., the set where the vector fields become collinear) and the heat cannot flow through the singularity. This is an interesting phenomenon since when approaching the singular set all Riemannian quantities explode, but geodesics are still well defined and can cross the singular set without singularities.

This phenomenon also appears in sub-Riemannian structures which are not equiregular, i.e., when the growth vector depends on the point. We show this fact by analyzing the Martinet case.

DOI : 10.5802/aif.2813
Classification : 53C17, 35P05, 58C40
Keywords: Grushin, Laplace-Beltrami operator, almost-Riemannian structures
Mot clés : Grushin, opérateur de Laplace-Beltrami, structures presque-Riemannienne
Boscain, Ugo 1 ; Laurent, Camille 2

1 CNRS, Centre de Mathématiques Appliquées, École Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France, and INRIA, Centre de Recherche Saclay, Team GECO.
2 CNRS, Centre de Mathématiques Appliquées, École Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France and CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005 Paris, France
@article{AIF_2013__63_5_1739_0,
     author = {Boscain, Ugo and Laurent, Camille},
     title = {The {Laplace-Beltrami} operator {in~almost-Riemannian} {Geometry}},
     journal = {Annales de l'Institut Fourier},
     pages = {1739--1770},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {63},
     number = {5},
     year = {2013},
     doi = {10.5802/aif.2813},
     zbl = {06284531},
     mrnumber = {3186507},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2813/}
}
TY  - JOUR
AU  - Boscain, Ugo
AU  - Laurent, Camille
TI  - The Laplace-Beltrami operator in almost-Riemannian Geometry
JO  - Annales de l'Institut Fourier
PY  - 2013
SP  - 1739
EP  - 1770
VL  - 63
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2813/
DO  - 10.5802/aif.2813
LA  - en
ID  - AIF_2013__63_5_1739_0
ER  - 
%0 Journal Article
%A Boscain, Ugo
%A Laurent, Camille
%T The Laplace-Beltrami operator in almost-Riemannian Geometry
%J Annales de l'Institut Fourier
%D 2013
%P 1739-1770
%V 63
%N 5
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2813/
%R 10.5802/aif.2813
%G en
%F AIF_2013__63_5_1739_0
Boscain, Ugo; Laurent, Camille. The Laplace-Beltrami operator in almost-Riemannian Geometry. Annales de l'Institut Fourier, Tome 63 (2013) no. 5, pp. 1739-1770. doi : 10.5802/aif.2813. http://www.numdam.org/articles/10.5802/aif.2813/

[1] Agrachev, A.; Barilari, D.; Boscain, U. Introduction to Riemannian and sub-Riemannian geometry (Lecture Notes) (http://people.sissa.it/agrachev/agrachev_files /notes.html.)

[2] Agrachev, A. A.; Boscain, U.; Charlot, G.; Ghezzi, R.; Sigalotti, M. Two-dimensional almost-Riemannian structures with tangency points, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 27 (2010) no. 3, pp. 793-807 | DOI | Numdam | MR | Zbl

[3] 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 | DOI | MR | Zbl

[4] Agrachev, Andrei; Boscain, Ugo; Sigalotti, Mario A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds, Discrete Contin. Dyn. Syst., Volume 20 (2008) no. 4, pp. 801-822 | DOI | MR | Zbl

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

[6] Barilari, Davide Trace heat kernel asymptotics in 3D contact sub-Riemannian geometry, Journal of Mathematical Science, To appear | Zbl

[7] Barilari, Davide; Boscain, Ugo; Neel, Robert Small time heat kernel asymptotics at the sub-Riemannian cut-locus, Journal of Differential Geometry, Volume 92 (2012) no. 3, pp. 373-416 | MR | Zbl

[8] Bellaïche, André The tangent space in sub-Riemannian geometry, Sub-Riemannian geometry (Progr. Math.), Volume 144, Birkhäuser, Basel, 1996, pp. 1-78 | MR | Zbl

[9] Bonnard, B.; Caillau, J.-B.; Sinclair, R.; Tanaka, M. Conjugate and cut loci of a two-sphere of revolution with application to optimal control, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 26 (2009) no. 4, pp. 1081-1098 | DOI | EuDML | Numdam | MR | Zbl

[10] Bonnard, Bernard; Caillau, Jean Baptiste Metrics with equatorial singularities on the sphere (Preprint 2011, HAL, vol. 00319299, pp. 1-29)

[11] Bonnard, Bernard; Charlot, Grégoire; Ghezzi, Roberta; Janin, Gabriel The Sphere and the Cut Locus at a Tangency Point in Two-Dimensional Almost-Riemannian Geometry, J. Dynam. Control Systems, Volume 17 (2011) no. 1, pp. 141-161 | DOI | MR | Zbl

[12] Bonnard, Bernard; Chyba, Monique Singular trajectories and their role in control theory, Mathématiques & Applications (Berlin) [Mathematics & Applications], 40, Springer-Verlag, Berlin, 2003 | MR | Zbl

[13] Boscain, U.; Chambrion, T.; Charlot, G. Nonisotropic 3-level quantum systems: complete solutions for minimum time and minimum energy, Discrete Contin. Dyn. Syst. Ser. B, Volume 5 (2005) no. 4, pp. 957-990 | DOI | MR | Zbl

[14] Boscain, U.; Charlot, G. Resonance of minimizers for n-level quantum systems with an arbitrary cost, ESAIM Control Optim. Calc. Var., Volume 10 (2004) no. 4, p. 593-614 (electronic) | DOI | Numdam | MR | Zbl

[15] Boscain, U.; Charlot, G.; Gauthier, J.-P.; Guérin, S.; Jauslin, H.-R. Optimal control in laser-induced population transfer for two- and three-level quantum systems, J. Math. Phys., Volume 43 (2002) no. 5, pp. 2107-2132 | DOI | MR | Zbl

[16] Boscain, U.; Charlot, G.; Ghezzi, R. Normal forms and invariants for 2-dimensional almost-Riemannian structures, Differential Geometry and its Applications, Volume 31 (2013) no. 1, pp. 41-62 | DOI | MR | Zbl

[17] Boscain, Ugo; Charlot, Gregoire; Ghezzi, Roberta; Sigalotti, Mario Lipschitz Classification of Two-Dimensional Almost-Riemannian Distances on Compact Oriented Surfaces, J. Geom. Anal., Volume 23 (2013) no. 1, pp. 438-455 | DOI | MR | Zbl

[18] Boscain, Ugo; Sigalotti, Mario High-order angles in almost-Riemannian geometry, Actes de Séminaire de Théorie Spectrale et Géométrie. Vol. 24. Année 2005–2006 (Sémin. Théor. Spectr. Géom.), Volume 25, Univ. Grenoble I, 2008, pp. 41-54 | Numdam | MR | Zbl

[19] Donnelly, H.; Garofalo, N. Schrödinger operators on manifolds, essential self-adjointness, and absence of eigenvalues, Journal of Geometric Analysis, Volume 7 (1997) no. 2, pp. 241-257 | DOI | MR | Zbl

[20] Franchi, Bruno; Lanconelli, Ermanno Une métrique associée à une classe d’opérateurs elliptiques dégénérés, Rend. Sem. Mat. Univ. Politec. Torino (1983) no. Special Issue, p. 105-114 (1984) Conference on linear partial and pseudodifferential operators (Torino, 1982) | MR | Zbl

[21] Grušin, V. V. A certain class of hypoelliptic operators, Mat. Sb. (N.S.), Volume 83 (125) (1970), pp. 456-473 | MR

[22] Gérard, P.; Grellier, S The Szegö cubic equation, Ann. Scient. Ec. Norm. Sup., Volume 43 (2010) no. 4, pp. 761-809 | Numdam | MR | Zbl

[23] Jean, Frédéric Uniform estimation of sub-Riemannian balls, J. Dynam. Control Systems, Volume 7 (2001) no. 4, pp. 473-500 | DOI | MR | Zbl

[24] Kalf, H.; Walter, J. Note on a paper of Simon on essentially self-adjoint Schrödinger operators with singular potentials, Archive for Rational Mechanics and Analysis, Volume 52 (1973) no. 3, pp. 258-260 | DOI | MR | Zbl

[25] Kato, T. Schrödinger operators with singular potentials, Israel Journal of Mathematics, Volume 13 (1972) no. 1, pp. 135-148 | DOI | MR | Zbl

[26] Léandre, Rémi Minoration en temps petit de la densité d’une diffusion dégénérée, J. Funct. Anal., Volume 74 (1987) no. 2, pp. 399-414 | DOI | MR | Zbl

[27] Maeda, M. Essential selfadjointness of Schrödinger operators with potentials singular along affine subspaces, Hiroshima Mathematical Journal, Volume 11 (1981) no. 2, pp. 275-283 | MR | Zbl

[28] Montgomery, Richard A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, 91, American Mathematical Society, Providence, RI, 2002 | MR | Zbl

[29] Neel, Robert The small-time asymptotics of the heat kernel at the cut locus, Comm. Anal. Geom., Volume 15 (2007) no. 4, pp. 845-890 http://projecteuclid.org/getRecord?id=euclid.cag/1208527887 | DOI | MR | Zbl

[30] Neel, Robert; Stroock, Daniel Analysis of the cut locus via the heat kernel, Surveys in differential geometry. Vol. IX (Surv. Differ. Geom., IX), Int. Press, Somerville, MA, 2004, pp. 337-349 | MR | Zbl

[31] Pontryagin, L. S.; Boltyanskiĭ, V. G.; Gamkrelidze, R. V.; Mishchenko, E. F. The Mathematical Theory of Optimal Processes, “Nauka”, Moscow, 1983 | MR | Zbl

[32] Reed, M.; Simon, B. Methods of modern mathematical physics, Academic press, 1980 | MR | Zbl

[33] Simon, B. Essential self-adjointness of Schrödinger operators with singular potentials, Archive for Rational Mechanics and Analysis, Volume 52 (1973) no. 1, pp. 44-48 | DOI | MR | Zbl

[34] Varadhan, S. R. S. On the behavior of the fundamental solution of the heat equation with variable coefficients, Comm. Pure Appl. Math., Volume 20 (1967), pp. 431-455 | DOI | MR | Zbl

[35] Vendittelli, Marilena; Oriolo, Giuseppe; Jean, Frédéric; Laumond, Jean-Paul Nonhomogeneous nilpotent approximations for nonholonomic systems with singularities, IEEE Trans. Automat. Control, Volume 49 (2004) no. 2, pp. 261-266 | DOI | MR

Cité par Sources :