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Colbois, Bruno
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Petites valeurs propres des p-formes différentielles et classe d'Euler des S1-fibrés
2000-12-21, Colbois, Bruno, Courtois, Gilles
Let M(n, a, d) be the set of compact oriented Riemannian manifolds (M, g) of dimension n whose sectional curvature K-g and diameter d(g) satisfy \K-g\ less than or equal to a and d(g) less than or equal to d. Let M(n, a, d, rho) be the subset of M(n, a, d) of those manifolds (M, g) such that the injectivity radius is greater than or equal to rho. if (M, g) is an element of M(n + 1, a, d) and (N, h) is an element of M(n, a', d') are sufficiently close in the sense of Gromov-Hausdorff, M is a circle bundle over N according to a theorem of K. Fukaya. When the Gromov-Hausdorff distance between (M, g) and (N, h) is small enough, we show that there exists m(p) - b(p)(N) + b(p-1) (N) - b(p)(M) small eigenvalues of the Laplacian acting on differential p-forms on M, 1 < p < n + 1, where b(p) denotes the p-th Betti number. We give uniform bounds of these eigenvalues depending on the Euler class of the circle bundle S-1 --> M --> N and the Gromov-Hausdorff distance between (M, g) and (N, h). (C) 2000 Editions scientifiques et medicales Elsevier SAS.
A note on the 1st nonzero eigenvalue of the laplacian acting on p-forms
1990, Colbois, Bruno, Courtois, Gilles
Sur la multiplicité de la première valeur propre de l'opérateur de Schrödinger avec champ magnétique sur la sphère S2
1998, Besson, Gérard, Colbois, Bruno, Courtois, Gilles
The purpose of this text is to study the first eigenvalue of Schrodinger operator with magnetic field on the 2-sphere and to show that its multiplicity can be arbitrarily high. We also show that this multiplicity is bounded in terms of the curvature of the corresponding connection. This answers a question asked by Y. Colin de Verdiere.
Les petites valeurs propres des variétés hyperboliques de dimension 3
1989-12-20, Colbois, Bruno, Courtois, Gilles
Convergence de variétés et convergence du spectre du Laplacien
1991-12-21, Colbois, Bruno, Courtois, Gilles
Les valeurs propres inférieures à 1/4 des surfaces de Riemann de petit rayon d'injectivité
1989, Colbois, Bruno, Courtois, Gilles