Colbois, Bruno

2009, Colbois, Bruno, Newberger, Florence, Verovic, Patrick

Given a closed hyperbolic Riemannian surface, the aim of the present paper is to describe an explicit construction of smooth deformations of the hyperbolic metric into Finsler metrics that are not Riemannian and whose properties are such that the classical Riemannian results about entropy rigidity, marked length spectrum rigidity and boundary rigidity all fail to extend to the Finsler category.

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.

1993, Buser, Peter, Colbois, Bruno, Dodziuk, Jozef

We investigate the structure of the spectrum near zero for the Laplace operator on a complete negatively curved Riemannian manifold M. If the manifold is compact and its sectional curvatures K satisfy 1 less-than-or-equal-to K < 0, we show that the smallest positive eigenvalue of the Laplacian is bounded below by a constant depending only on the volume of M. Our result for a complete manifold of finite volume with sectional curvatures pinched between -a2 and -1 asserts that the number of eigenvalues of the Laplacian between 0 and (n -1)2/4 is bounded by a constant multiple of the volume of the manifold with the constant depending on a and the dimension only.