Faculté des sciences

Bornes supérieures pour les valeurs propres des opérateurs naturels sur des variétés riemanniennes compactes

Hassannezhad, Asma ; Colbois, Bruno (Dir.) ; El Soufi, Ahmad (Codir.) ; Ranjbar-Motlagh, Alireza (Codir.)

Thèse de doctorat : Université de Neuchâtel, 2012.

The purpose of this thesis is to find upper bounds for the eigenvalues of natural operators acting on functions on a compact Riemannian manifold (M, g) 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... Mehr

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    Summary
    The purpose of this thesis is to find upper bounds for the eigenvalues of natural operators acting on functions on a compact Riemannian manifold (M, g) 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.
    The second aspect is to study the interplay of the extrinsic geometry and eigenvalues of the Laplace-Beltrami operator acting on compact submanifolds of RN and of CPN. We investigate an extrinsic invariant called the intersection index studied by Colbois, Dryden and El Soufi. For compact submanifolds of RN, we extend their results and obtain upper bounds which are stable under small perturbation. For compact submanifolds of CPN we obtain an upper bound depending only on the degree of submanifolds and which is sharp for the first eigenvalue.
    As a further application of the introduced method, we obtain an upper bound for the eigenvalues of the Steklov problem in a domain with C1 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.