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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
    Spectrum of the Laplacian with weights
    (2019-3-4)
    Colbois, Bruno 
    ;
    El Soufi, Ahmad
    Given a compact Riemannian manifold $(M,g)$ and two positive functions $\rho$ and $\sigma$, we are interested in the eigenvalues of the Dirichlet energy functional weighted by $\sigma$, with respect to the $L^2$ inner product weighted by $\rho$. Under some regularity conditions on $\rho$ and $\sigma$, these eigenvalues are those of the operator $-\rho^{-1} \mbox{div}(\sigma \nabla u)$ with Neumann conditions on the boundary if $\partial M\ne \emptyset$. We investigate the effect of the weights on eigenvalues and discuss the existence of lower and upper bounds under the condition that the total mass is preserved.
  • 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
    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).
  • Publication
    Métadonnées seulement
    Eigenvalues of the laplacian acting on p-forms and metric conformal deformations
    (2006)
    Colbois, Bruno 
    ;
    El Soufi, Ahmad
    Let (M, g) be a compact connected orientable Riemannian manifold of dimension n >= 4 and let lambda(k,p)(g) be the k-th positive eigenvalue of the Laplacian. Delta g,p = dd* + d* d acting on differential forms of degree p on M. We prove that the metric g can be conformally deformed to a metric g', having the same volume as g, with arbitrarily large lambda 1, p(g') for all p is an element of [2,n-2]. Note that for the other values of p, that is p = 0, 1, n-1 and n, one can deduce from the literature that, for all k > 0, the k-th eigenvalue lambda(k,p) is uniformly bounded on any conformal class of metrics of fixed volume on M. For p = 1, we show that, for any positive integer N, there exists a metric g(N) conformal to g such that, for all k
  • Publication
    Métadonnées seulement
    Extremal eigenvalues of the Laplacian in a conformal class of metrics: The 'conformal spectrum'
    (2003-12-21)
    Colbois, Bruno 
    ;
    El Soufi, Ahmad
    Let M be a compact connected manifold of dimension n endowed with a conformal class C of Riemannian metrics of volume one. For any integer k greater than or equal to 0, we consider the conformal invariant.c k( C) defined as the supremum of the k-th eigenvalue lambda(k)(g) of the Laplace-Beltrami operator Delta(g), where g runs over C. First, we give a sharp universal lower bound for lambda(k)(c)(C) extending to all k a result obtained by Friedlander and Nadirashvili for k = 1. Then, we show that the sequence {lambda(k)(c)(C)}, that we call 'conformal spectrum', is strictly increasing and satisfies, For Allk greater than or equal to 0, lambda(k+1)(c)(C)(n/2)-lambda(k)(c)(C)(n/2) greater than or equal to n(n/2) omega(n), where omega(n) is the volume of the n-dimensional standard sphere. When M is an orientable surface of genus gamma, we also consider the supremum zeta(k)(top) (gamma) of lambda(k)(g) over the set of all the area one Riemannian metrics on M, and study the behavior of lambda(k)(top)(gamma) in terms of gamma.