The Laplace-Beltrami operator in almost-Riemannian Geometry

Boscain, Ugo; Laurent, Camille

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.

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