Bornes supérieures pour les valeurs propres des opérateurs naturels sur des variétés riemanniennes compactes
Author(s)
Hassannezhad, Asma
Editor(s)
Colbois, Bruno
Chaire de géométrie
El Soufi, Ahmad
Ranjbar-Motlagh, Alireza
Date issued
2012
Subjects
Riemann, Variétés de Schrödinger, Opérateur de Laplacien Valeurs propres Opérateur de Laplace operateur de Schrödinger opérateur de Laplace Barky-Emery valeurs propres borne supérieure volume confrome minimal nombre d'intersection moyenne Laplace-Beltrami operator Schrödinger operator Bakry-Emery Laplace operator eigenvalue upper bound min-conformal volume mean intersection index
Abstract
The purpose of this thesis is to find upper bounds for the eigenvalues of natural operators acting on functions on a compact Riemannian manifold (<i>M</i>, <i>g</i>) such as the Laplace-Beltrami operator and Laplace-type operators. In the case of the Laplace-Beltrami operator, two aspects are investigated: The first aspect is to study relationships between the intrinsic geometry and eigenvalues of the Laplace-Beltrami operator. In this regard, we obtain upper bounds depending only on the dimension and a conformal invariant called min-conformal volume. Asymptotically, these bounds are consistent with the Weyl law. They improve previous results by Korevaar and Yang and Yau. The proof relies on the construction of a suitable family of disjoint domains providing supports for a family of test functions. This method is powerful and interesting in itself. <br> The second aspect is to study the interplay of the extrinsic geometry and eigenvalues of the Laplace-Beltrami operator acting on compact submanifolds of <i>R</i><sup>N</sup> and of <i>CP</i><sup>N</sup>. We investigate an extrinsic invariant called the intersection index studied by Colbois, Dryden and El Soufi. For compact submanifolds of <i>R</i><sup>N</sup>, we extend their results and obtain upper bounds which are stable under small perturbation. For compact submanifolds of <i>CP</i><sup>N</sup> we obtain an upper bound depending only on the degree of submanifolds and which is sharp for the first eigenvalue. <br> As a further application of the introduced method, we obtain an upper bound for the eigenvalues of the Steklov problem in a domain with <i>C</i><sup>1</sup> boundary in a complete Riemannian manifold in terms of the isoperimetric ratio of the domain and the min-conformal volume. A modification of our method also leads to have upper bounds for the eigenvalues of Schrödinger operators in terms of the min-conformal volume and integral quantity of the potential. As another application of our method, we obtain upper bounds for the eigenvalues of the Bakry-Emery Laplace operator depending on conformal invariants and properties of the weighted function.
Notes
Thèse de doctorat : Université de Neuchâtel, 2012
Publication type
doctoral thesis
File(s)