# 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

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topBoscain, 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 -

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## Citations in EuDML Documents

top- Davide Barilari, Luca Rizzi, A Formula for Popp’s Volume in Sub-Riemannian Geometry
- Davide Barilari, Ugo Boscain, Grégoire Charlot, Robert W. Neel, Asymptotiques en temps petit du noyau de la chaleur des métriques riemanniennes et sous-riemanniennes
- Yves Colin de Verdière, Luc Hillairet, Emmanuel Trélat, Quantum ergodicity and quantum limits for sub-Riemannian Laplacians

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