Colbois, Bruno

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.

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.

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.

2003, Colbois, Bruno, Matei, Ana-Maria

We study the relationship between the first eigenvalue of the Laplacian and Cheeger constant when the Cheeger constant converges to zero, in the case of compact Riemannian manifolds and of finite graphs. (C) 2003 Elsevier B.V. All rights reserved.

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