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Colbois, Bruno

Nom
Colbois, Bruno
Affiliation principale
Site web
Fonction
Professeur ordinaire
Email
Bruno.Colbois@unine.ch
Identifiants
https://libra.unine.ch/handle/123456789/456
_
0000-0002-8514-4315

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  • Publication
    Accès libre
    Inégalités géométriques pour des valeurs propres de Steklov de graphes et de surfaces
    (2023)
    Perrin, Hélène 
    ;
    Colbois, Bruno 
    Cette thèse est consacrée à l’obtention d’inégalités géométriques pour des valeurs propres de Steklov de variétés riemanniennes de dimension 2 et de graphes. Les résultats obtenus concernent différentes situations. D’un côté, je m’intéresse à la géométrie de la première valeur propre non nulle de Steklov σ1 d’un graphe à bord. Pour cette valeur propre, je donne une borne inférieure qui dépend d’une borne supérieure sur le diamètre extrinsèque du bord et d’une borne supérieure sur le nombre de sommets du bord. Un autre résultat est une borne supérieure pour certains sous-graphes d’un graphe de Cayley à croissance polynomiale, qui montre en particulier que σ1 tend vers 0 lorsque le nombre de sommets du sous-graphe tend vers l’infini et généralise ainsi un résultat de Han et Hua obtenu pour des sous-graphes de Zn. Un deuxième but de la thèse est d’obtenir des bornes inférieures pour la première valeur propre non nulle de Steklov σ1 d’une variété riemannienne M dont le bord a plusieurs composantes connexes. Dans ce cas, la géométrie de M loin du bord peut avoir une forte influence sur σ1. Afin de préciser la forme de cette relation on étudie les variétés riemanniennes dont le bord a un voisinage cylindrique. En dimension 2, en supposant que la courbure de Gauss de M est bornée inférieurement, je donne une borne inférieure qui dépend d’une borne supérieure sur le diamètre extrinsèque du bord, d’une borne supérieure sur la longueur du bord et d’une borne inférieure sur la rayon d’injectivité des points d’un certain sous-ensemble de M. Finalement, je donne des bornes inférieure et supérieure pour les premières valeurs propres de Steklov d’une surface hyperbolique à bord géodésique en fonction de la longueur de certaines familles de géodésiques qui séparent le bord. Ce résultat est similaire à un résultat classique de Schoen, Wolpert et Yau pour les valeurspropres du laplacien d’une surface hyperbolique fermée. Abstract The aim of this thesis is to obtain geometric inequalities for Steklov eigenvalues of 2-dimensional Riemannian manifolds and graphs. The results obtained relate to different situations. On the one hand, our interest focuses on the geometry of the first non-zero Steklov eigenvalue σ1 of a graph with boundary. For this eigenvalue, we give a lower bound which depends on an upper bound on the extrinsic diameter of the boundary and on an upper bound on the number of vertices of the boundary. Another result is an upper bound for some subgraphs of a Cayley graph with polynomial growth, which shows in particular that σ1 tends to 0 when the number of vertices of the subgraph tends to infinity and thus generalizes a result of Han and Hua obtained for subgraphs of Zn. A second goal of the thesis is to obtain lower bounds for the first non-zero Steklov eigenvalue σ1 of a Riemannian manifold M whose boundary has several connected components. In this case, the geometry of M far from the boundary can have a strong influence on σ1. In order to specify the form of this relation we study Riemannian manifolds whose boundary has a cylindrical neighborhood. In dimension 2, assuming that the Gaussian curvature of M is bounded below, we give a lower bound which depends on an upper bound on the extrinsic diameter of the boundary, an upper bound on the length of the boundary and a lower bound on the radius of injectivity at the points of a certain subset of M. Finally, we give lower and upper bounds for the first Steklov eigenvalues of hyperbolic surfaces with geodesic boundary, which depend on the length of some families of geodesics that separate the boundary.This result is similar to a classical result of Schoen, Wolpert and Yau for Laplace eigenvalues of a closed hyperbolic surface.
  • Publication
    Accès libre
    Bornes supérieures pour les valeurs propres des opérateurs naturels sur des variétés riemanniennes compactes
    (2012)
    Hassannezhad, Asma
    ;
    Colbois, Bruno 
    ;
    El Soufi, Ahmad
    ;
    Ranjbar-Motlagh, Alireza
    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.
  • Publication
    Métadonnées seulement
    Bounding the eigenvalues of the Laplace-Beltrami operator on compact submanifolds
    (2010-1-21)
    Colbois, Bruno 
    ;
    Dryden, Emily B
    ;
    El Soufi, Ahmad
    We give upper bounds for the eigenvalues of the La-place-Beltrami operator of a compact m-dimensional submanifold M of R^{m+p}. Besides the dimension and the volume of the submanifold and the order of the eigenvalue, these bounds depend on either the maximal number of intersection points of M with a p-plane in a generic position (transverse to M), or an invariant which measures the concentration of the volume of M in R^{m+p}. These bounds are asymptotically optimal in the sense of the Weyl law. On the other hand, we show that even for hypersurfaces (i.e., when p=1), the first positive eigenvalue cannot be controlled only in terms of the volume, the dimension and (for m>2) the differential structure.
  • Publication
    Métadonnées seulement
    Eigenvalues estimate for the Neumann problem of a bounded domain
    (2008-12-21)
    Colbois, Bruno 
    ;
    Maerten, Daniel
    In this note, we investigate upper bounds of the Neumann eigenvalue problem for the Laplacian of a domain Omega in a given complete ( not compact a priori) Riemannian manifold ( M, g). For this, we use test functions for the Rayleigh quotient subordinated to a family of open sets constructed in a general metric way, interesting for itself. As applications, we prove that if the Ricci curvature of ( M, g) is bounded below Ric(g) >= -( n - 1) a(2), a >= 0, then there exist constants A(n) > 0, B-n > 0 only depending on the dimension, such that lambda(k)(Omega)
  • Publication
    Métadonnées seulement
    Extremal g-invariant eigenvalues of the Laplacian of g-invariant metrics
    (2008-12-21)
    Colbois, Bruno 
    ;
    Dryden, Emily B
    ;
    El Soufi, Ahmad
    The study of extremal properties of the spectrum often involves restricting the metrics under consideration. Motivated by the work of Abreu and Freitas in the case of the sphere S-2 endowed with S-1-invariant metrics, we consider the subsequence lambda(G)(k) of the spectrum of a Riemannian manifold M which corresponds to metrics and functions invariant under the action of a compact Lie group G. If. G has dimension at least 1, we show that the functional lambda(G)(k) admits no extremal metric under volume-preserving G-invariant deforma- tions. If, moreover, M has dimension at least three, then the functional lambda(G)(k) is unbounded when restricted to any conformal class of G-invariant metrics of fixed volume. As a special case of this, we can consider the standard 0(n)-action on S-n; however, if we also require the metric to be induced by an embedding of S-n in Rn+1, we get an optimal upper bound on lambda(G)(k).