The Laplace-Beltrami operator in almost-Riemannian Geometry

Ugo Boscain[1]; Camille Laurent[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

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 5, page 1739-1770
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Boscain, Ugo, and Laurent, Camille. "The Laplace-Beltrami operator in almost-Riemannian Geometry." Annales de l’institut Fourier 63.5 (2013): 1739-1770. <http://eudml.org/doc/275451>.

@article{Boscain2013,
abstract = {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.},
affiliation = {CNRS, Centre de Mathématiques Appliquées, École Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France, and INRIA, Centre de Recherche Saclay, Team GECO.; 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},
author = {Boscain, Ugo, Laurent, Camille},
journal = {Annales de l’institut Fourier},
keywords = {Grushin; Laplace-Beltrami operator; almost-Riemannian structures; almost Riemannian structures; Grushin points; PDEs and singularities; geodesics; area elements},
language = {eng},
number = {5},
pages = {1739-1770},
publisher = {Association des Annales de l’institut Fourier},
title = {The Laplace-Beltrami operator in almost-Riemannian Geometry},
url = {http://eudml.org/doc/275451},
volume = {63},
year = {2013},
}

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
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 5
SP - 1739
EP - 1770
AB - 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.
LA - eng
KW - Grushin; Laplace-Beltrami operator; almost-Riemannian structures; almost Riemannian structures; Grushin points; PDEs and singularities; geodesics; area elements
UR - http://eudml.org/doc/275451
ER -

References

top
  1. A. Agrachev, D. Barilari, U. Boscain, Introduction to Riemannian and sub-Riemannian geometry (Lecture Notes) Zbl1236.53030
  2. A. A. Agrachev, U. Boscain, G. Charlot, R. Ghezzi, M. Sigalotti, Two-dimensional almost-Riemannian structures with tangency points, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), 793-807 Zbl1192.53029MR2629880
  3. Andrei Agrachev, Ugo Boscain, Jean-Paul Gauthier, Francesco Rossi, The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups, J. Funct. Anal. 256 (2009), 2621-2655 Zbl1165.58012MR2502528
  4. Andrei Agrachev, Ugo Boscain, Mario Sigalotti, A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds, Discrete Contin. Dyn. Syst. 20 (2008), 801-822 Zbl1198.49041MR2379474
  5. Andrei A. Agrachev, Yuri L. Sachkov, Control theory from the geometric viewpoint, 87 (2004), Springer-Verlag, Berlin Zbl1062.93001MR2062547
  6. Davide Barilari, Trace heat kernel asymptotics in 3D contact sub-Riemannian geometry, Journal of Mathematical Science, To appear Zbl1294.58004
  7. Davide Barilari, Ugo Boscain, Robert Neel, Small time heat kernel asymptotics at the sub-Riemannian cut-locus, Journal of Differential Geometry 92 (2012), 373-416 Zbl1270.53066MR3005058
  8. André Bellaïche, The tangent space in sub-Riemannian geometry, Sub-Riemannian geometry 144 (1996), 1-78, Birkhäuser, Basel Zbl0862.53031MR1421822
  9. B. Bonnard, J.-B. Caillau, R. Sinclair, M. Tanaka, Conjugate and cut loci of a two-sphere of revolution with application to optimal control, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 1081-1098 Zbl1184.53036MR2542715
  10. Bernard Bonnard, Jean Baptiste Caillau, Metrics with equatorial singularities on the sphere 
  11. Bernard Bonnard, Grégoire Charlot, Roberta Ghezzi, Gabriel Janin, The Sphere and the Cut Locus at a Tangency Point in Two-Dimensional Almost-Riemannian Geometry, J. Dynam. Control Systems 17 (2011), 141-161 Zbl1209.53014MR2765542
  12. Bernard Bonnard, Monique Chyba, Singular trajectories and their role in control theory, 40 (2003), Springer-Verlag, Berlin Zbl1022.93003MR1996448
  13. U. Boscain, T. Chambrion, G. Charlot, Nonisotropic 3-level quantum systems: complete solutions for minimum time and minimum energy, Discrete Contin. Dyn. Syst. Ser. B 5 (2005), 957-990 Zbl1084.81083MR2170218
  14. U. Boscain, G. Charlot, Resonance of minimizers for n -level quantum systems with an arbitrary cost, ESAIM Control Optim. Calc. Var. 10 (2004), 593-614 (electronic) Zbl1072.49002MR2111082
  15. U. Boscain, G. Charlot, J.-P. Gauthier, S. Guérin, H.-R. Jauslin, Optimal control in laser-induced population transfer for two- and three-level quantum systems, J. Math. Phys. 43 (2002), 2107-2132 Zbl1059.81195MR1893663
  16. U. Boscain, G. Charlot, R. Ghezzi, Normal forms and invariants for 2-dimensional almost-Riemannian structures, Differential Geometry and its Applications 31 (2013), 41-62 Zbl1260.53063MR3010077
  17. Ugo Boscain, Gregoire Charlot, Roberta Ghezzi, Mario Sigalotti, Lipschitz Classification of Two-Dimensional Almost-Riemannian Distances on Compact Oriented Surfaces, J. Geom. Anal. 23 (2013), 438-455 Zbl1259.53031MR3010287
  18. Ugo Boscain, Mario Sigalotti, High-order angles in almost-Riemannian geometry, Actes de Séminaire de Théorie Spectrale et Géométrie. Vol. 24. Année 2005–2006 25 (2008), 41-54, Univ. Grenoble I Zbl1159.53320MR2478807
  19. H. Donnelly, N. Garofalo, Schrödinger operators on manifolds, essential self-adjointness, and absence of eigenvalues, Journal of Geometric Analysis 7 (1997), 241-257 Zbl0912.58043MR1646768
  20. Bruno Franchi, Ermanno Lanconelli, Une métrique associée à une classe d’opérateurs elliptiques dégénérés, Rend. Sem. Mat. Univ. Politec. Torino (1983), 105-114 (1984) Zbl0553.35033MR745979
  21. V. V. Grušin, A certain class of hypoelliptic operators, Mat. Sb. (N.S.) 83 (125) (1970), 456-473 MR279436
  22. P. Gérard, S Grellier, The Szegö cubic equation, Ann. Scient. Ec. Norm. Sup. 43 (2010), 761-809 Zbl1228.35225MR2721876
  23. Frédéric Jean, Uniform estimation of sub-Riemannian balls, J. Dynam. Control Systems 7 (2001), 473-500 Zbl1029.53039MR1854033
  24. H. Kalf, J. Walter, Note on a paper of Simon on essentially self-adjoint Schrödinger operators with singular potentials, Archive for Rational Mechanics and Analysis 52 (1973), 258-260 Zbl0277.47008MR338549
  25. T. Kato, Schrödinger operators with singular potentials, Israel Journal of Mathematics 13 (1972), 135-148 Zbl0246.35025MR333833
  26. Rémi Léandre, Minoration en temps petit de la densité d’une diffusion dégénérée, J. Funct. Anal. 74 (1987), 399-414 Zbl0637.58034MR904825
  27. M. Maeda, Essential selfadjointness of Schrödinger operators with potentials singular along affine subspaces, Hiroshima Mathematical Journal 11 (1981), 275-283 Zbl0513.35025MR620538
  28. Richard Montgomery, A tour of subriemannian geometries, their geodesics and applications, 91 (2002), American Mathematical Society, Providence, RI Zbl1044.53022MR1867362
  29. Robert Neel, The small-time asymptotics of the heat kernel at the cut locus, Comm. Anal. Geom. 15 (2007), 845-890 Zbl1154.58020MR2395259
  30. Robert Neel, Daniel Stroock, Analysis of the cut locus via the heat kernel, Surveys in differential geometry. Vol. IX (2004), 337-349, Int. Press, Somerville, MA Zbl1081.58013MR2195412
  31. L. S. Pontryagin, V. G. Boltyanskiĭ, R. V. Gamkrelidze, E. F. Mishchenko, The Mathematical Theory of Optimal Processes, (1983), “Nauka”, Moscow Zbl0516.49001MR719372
  32. M. Reed, B. Simon, Methods of modern mathematical physics, (1980), Academic press Zbl0459.46001MR751959
  33. B. Simon, Essential self-adjointness of Schrödinger operators with singular potentials, Archive for Rational Mechanics and Analysis 52 (1973), 44-48 Zbl0277.47007MR338548
  34. S. R. S. Varadhan, On the behavior of the fundamental solution of the heat equation with variable coefficients, Comm. Pure Appl. Math. 20 (1967), 431-455 Zbl0155.16503MR208191
  35. Marilena Vendittelli, Giuseppe Oriolo, Frédéric Jean, Jean-Paul Laumond, Nonhomogeneous nilpotent approximations for nonholonomic systems with singularities, IEEE Trans. Automat. Control 49 (2004), 261-266 MR2034349

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