Spectral
theory of the Laplacian
In our thesis and in [10] we studied the fine spectral structure of the Laplacian on a complete manifold. We proved, in an invariant way, certain estimates from quantum mechanical scattering theory. Using the theory of H-smooth operators of T. Kato we proved that the spectrum of the Laplacian on a complete manifold M is absolutely continuous with respect to the Lebesgue measure (a much more refined property than absence of eigenvalues), provided the manifold carries certain convex functions. In [5] we proved a vanishing theorem for L^2 cohomology of differential forms. It states that if the curvatures of a Hadamard manifold are sufficiently pinched then there are non non-trivial square integrable harmonic forms outside the middle dimension. As in our previous work on spectral theory of the Laplacian, convexity also plays an important role in [5]. In [18] we apply the Hilbert space methods of our thesis to the study of the topological pressure of the cocycle given by the (bi)-Laplacian of the Busemann functions on the universal cover of a compact manifold of negative curvature. The result is interesting because the Laplacian of the Busemann functions is essentially the integrand in the famous Pesin formula for the entropy of the geodesic flow. In the preprint [23] we introduce the notion of (spectrally) \it stable \rm manifolds. These are complete manifolds (M,g) for which the unitary equivalence class of the L^2 Laplacian remains constant as $g$ varies in a neighborhood, relative to a suitable topology in the space of metrics. This concept of stability is natural, even compelling. For instance, no compact manifold is stable, but Euclidean flat space is stable. In [23] we prove that hyperbolic n - space is stable for all n>1.
Related Publications
[5] F. Xavier (with H. Donnelly), On the differential form spectrum of negatively curved manifolds, American Journal of Mathematics, 106, 169-187, (1984).
[10] F. Xavier, Convexity and absolute continuity of the Laplace-Beltrami operator, Math. Ann., 282, 579-585, (l988).
[18] F. Xavier (with V. Nitica) Schrodinger operators and topological pressure on manifolds of negative curvature, Proc. Symposia of AMS, Smooth ergodic theory and Applications, edited by Katok, vol. 69, 791-796 (2001)
[23] F. Xavier (with H. Donnelly) Spectral stability of hyperbolic spaces (Math Z., to appear)