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, p. 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 : https://doi.org/10.5802/aif.2813
Classification:  53C17,  35P05,  58C40
Mots clés: Grushin, opérateur de Laplace-Beltrami, structures presque-Riemannienne
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {63},
     number = {5},
     year = {2013},
     pages = {1739-1770},
     doi = {10.5802/aif.2813},
     mrnumber = {3186507},
     zbl = {06284531},
     language = {en},
     url = {http://http://www.numdam.org/item/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/item/AIF_2013__63_5_1739_0/

[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, Tome 27 (2010) no. 3, pp. 793-807 | Article | Numdam | MR 2629880 | Zbl 1192.53029

[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., Tome 256 (2009) no. 8, pp. 2621-2655 | Article | MR 2502528 | Zbl 1165.58012

[4] Agrachev, Andrei; Boscain, Ugo; Sigalotti, Mario A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds, Discrete Contin. Dyn. Syst., Tome 20 (2008) no. 4, pp. 801-822 | Article | MR 2379474 | Zbl 1198.49041

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

[6] Barilari, Davide Trace heat kernel asymptotics in 3D contact sub-Riemannian geometry, Journal of Mathematical Science, To appear | Zbl 1294.58004

[7] Barilari, Davide; Boscain, Ugo; Neel, Robert Small time heat kernel asymptotics at the sub-Riemannian cut-locus, Journal of Differential Geometry, Tome 92 (2012) no. 3, pp. 373-416 | MR 3005058 | Zbl 1270.53066

[8] Bellaïche, André The tangent space in sub-Riemannian geometry, Sub-Riemannian geometry, Birkhäuser, Basel (Progr. Math.) Tome 144 (1996), pp. 1-78 | MR 1421822 | Zbl 0862.53031

[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, Tome 26 (2009) no. 4, pp. 1081-1098 | Article | Numdam | MR 2542715 | Zbl 1184.53036

[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, Tome 17 (2011) no. 1, pp. 141-161 | Article | MR 2765542 | Zbl 1209.53014

[12] Bonnard, Bernard; Chyba, Monique Singular trajectories and their role in control theory, Springer-Verlag, Berlin, Mathématiques & Applications (Berlin) [Mathematics & Applications], Tome 40 (2003) | MR 1996448 | Zbl 1022.93003

[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, Tome 5 (2005) no. 4, pp. 957-990 | Article | MR 2170218 | Zbl 1084.81083

[14] Boscain, U.; Charlot, G. Resonance of minimizers for n-level quantum systems with an arbitrary cost, ESAIM Control Optim. Calc. Var., Tome 10 (2004) no. 4, p. 593-614 (electronic) | Article | Numdam | MR 2111082 | Zbl 1072.49002

[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., Tome 43 (2002) no. 5, pp. 2107-2132 | Article | MR 1893663 | Zbl 1059.81195

[16] Boscain, U.; Charlot, G.; Ghezzi, R. Normal forms and invariants for 2-dimensional almost-Riemannian structures, Differential Geometry and its Applications, Tome 31 (2013) no. 1, pp. 41-62 | Article | MR 3010077 | Zbl 1260.53063

[17] Boscain, Ugo; Charlot, Gregoire; Ghezzi, Roberta; Sigalotti, Mario Lipschitz Classification of Two-Dimensional Almost-Riemannian Distances on Compact Oriented Surfaces, J. Geom. Anal., Tome 23 (2013) no. 1, pp. 438-455 | Article | MR 3010287 | Zbl 1259.53031

[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, Univ. Grenoble I (Sémin. Théor. Spectr. Géom.) Tome 25 (2008), pp. 41-54 | Numdam | MR 2478807 | Zbl 1159.53320

[19] Donnelly, H.; Garofalo, N. Schrödinger operators on manifolds, essential self-adjointness, and absence of eigenvalues, Journal of Geometric Analysis, Tome 7 (1997) no. 2, pp. 241-257 | Article | MR 1646768 | Zbl 0912.58043

[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 745979 | Zbl 0553.35033

[21] Grušin, V. V. A certain class of hypoelliptic operators, Mat. Sb. (N.S.), Tome 83 (125) (1970), pp. 456-473 | MR 279436

[22] Gérard, P.; Grellier, S The Szegö cubic equation, Ann. Scient. Ec. Norm. Sup., Tome 43 (2010) no. 4, pp. 761-809 | Numdam | MR 2721876 | Zbl 1228.35225

[23] Jean, Frédéric Uniform estimation of sub-Riemannian balls, J. Dynam. Control Systems, Tome 7 (2001) no. 4, pp. 473-500 | Article | MR 1854033 | Zbl 1029.53039

[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, Tome 52 (1973) no. 3, pp. 258-260 | Article | MR 338549 | Zbl 0277.47008

[25] Kato, T. Schrödinger operators with singular potentials, Israel Journal of Mathematics, Tome 13 (1972) no. 1, pp. 135-148 | Article | MR 333833 | Zbl 0246.35025

[26] Léandre, Rémi Minoration en temps petit de la densité d’une diffusion dégénérée, J. Funct. Anal., Tome 74 (1987) no. 2, pp. 399-414 | Article | MR 904825 | Zbl 0637.58034

[27] Maeda, M. Essential selfadjointness of Schrödinger operators with potentials singular along affine subspaces, Hiroshima Mathematical Journal, Tome 11 (1981) no. 2, pp. 275-283 | MR 620538 | Zbl 0513.35025

[28] Montgomery, Richard A tour of subriemannian geometries, their geodesics and applications, American Mathematical Society, Providence, RI, Mathematical Surveys and Monographs, Tome 91 (2002) | MR 1867362 | Zbl 1044.53022

[29] Neel, Robert The small-time asymptotics of the heat kernel at the cut locus, Comm. Anal. Geom., Tome 15 (2007) no. 4, pp. 845-890 | Article | MR 2395259 | Zbl 1154.58020

[30] Neel, Robert; Stroock, Daniel Analysis of the cut locus via the heat kernel, Surveys in differential geometry. Vol. IX, Int. Press, Somerville, MA (Surv. Differ. Geom., IX) (2004), pp. 337-349 | MR 2195412 | Zbl 1081.58013

[31] Pontryagin, L. S.; Boltyanskiĭ, V. G.; Gamkrelidze, R. V.; Mishchenko, E. F. The Mathematical Theory of Optimal Processes, “Nauka”, Moscow (1983) | MR 719372 | Zbl 0516.49001

[32] Reed, M.; Simon, B. Methods of modern mathematical physics, Academic press (1980) | MR 751959 | Zbl 0459.46001

[33] Simon, B. Essential self-adjointness of Schrödinger operators with singular potentials, Archive for Rational Mechanics and Analysis, Tome 52 (1973) no. 1, pp. 44-48 | Article | MR 338548 | Zbl 0277.47007

[34] Varadhan, S. R. S. On the behavior of the fundamental solution of the heat equation with variable coefficients, Comm. Pure Appl. Math., Tome 20 (1967), pp. 431-455 | Article | MR 208191 | Zbl 0155.16503

[35] Vendittelli, Marilena; Oriolo, Giuseppe; Jean, Frédéric; Laumond, Jean-Paul Nonhomogeneous nilpotent approximations for nonholonomic systems with singularities, IEEE Trans. Automat. Control, Tome 49 (2004) no. 2, pp. 261-266 | Article | MR 2034349