Small eigenvalues and Euler class of circle bundles

Let M(n, a, d) be the set of compact oriented Riemannian manifolds (M, g) o f 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, r ho) be the subset of M(n, a, d) of those manifolds (M, g) such that the inj ectivity 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 c lose in the sense of Gromov-Hausdorff, M is a circle bundle over N accordin g 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 different ial 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 betwee n (M, g) and (N, h). (C) 2000 Editions scientifiques et medicales Elsevier SAS.