Extension of Bochner-Lichnérowicz formula on spheres

Dominique Bakry (1) , Abdellatif Ben-Taleb

1 IMT - Institut de Mathématiques de Toulouse UMR5219

Given a second order differential operator L, we define the vector space of ”intrinsic bilinear operators” associated with it. They are constructed only from the operator L itself and the algebra structure given by the product of functions. When the operator is symmetric with respect to some positive measure, every positive quadratic form in this space provides information on the spectrum of the operator. The positiveness of a form relies only on the local structure of the operator. The purpose of this paper to construct a sequence (Rk) of positive intrinsic quadratic forms on spheres (in this case, L is the Laplace-Beltrami operator) which carry all the information on the spectrum. This extends to the full spectrum the Bochner-Lichn´erowicz formula which gives on the sphere a sharp lower bound on the first non-zero eigenvalue. An extension of this property is given for a family of operators which extends the ultraspherical operator on the real line.

Mots clés: Bochner-Lichnérowicz formula; spectral analysis; Laplace-Beltrami operators; spherical laplacians

Domaines: Géométrie différentielle [math.DG]; Probabilités [math.PR]

Dominique Bakry, Abdellatif Ben-Taleb. Extension of Bochner-Lichnérowicz formula on spheres. Annales de la Faculté des Sciences de Toulouse. Mathématiques., 2005, 6, 14 (2), pp.161-183. ⟨hal-00353948⟩