Nelia Charalambous
Ph.D. (2004) Cornell University

First Position

Visiting assistant professor, Cornell University, Department of Mathematics

Dissertation

On the L^p Independence of the Spectrum of the Hodge Laplacian and Logarithmic Sobolev Inequalities on Non-Compact manifolds

Advisor: José Escobar
Research Area:Analytic Geometry, PDEs on Manifolds

Abstract: The central aim of this thesis is the study of the spectrum and Heat kernel bounds for the Hodge Laplacian on differential forms of any order k in the Banach Space L^p. The underlying space is a C^∞-smooth open manifold M^N, not necessarily compact, on which Ricci Curvature is bounded below, and has uniformly subexponential volume growth. It will be demonstrated that on such a space the L^p spectrum of the Hodge Laplacian on differential k-forms is independent of p for 1 ≤ p ≤ ∞, when the Weitzenböck Tensor on k-forms has a lower bound as well. It follows as a Corollary that the isolated eigenvalues of finite multiplicity are also L^p independent. By considering the L^p spectra of the Hodge Laplacian on the hyperbolic space H^{N+1} we conclude that the subexponential volume growth condition is necessary in the case of one-forms.

The proof of the L^p independence result, relies on finding a Gaussian type upper bound for the heat kernel of the Hodge Laplacian. For obtaining the Gaussian bound we will use Kato's inequality. It will be shown that similar bounds can be also obtained via logarithmic Sobolev inequalities by assuming a uniform lower bound on the volume of balls of radius one instead of a uniform subexponential volume growth.

Finally, as an application, we will show that the spectrum of the Laplacian on one-forms has no gaps on manifolds with a pole and on manifolds that are in a warped product form. This will be done under weaker curvature restrictions than what have been used previously; it will be achieved by finding the L^1 spectrum of the Laplacian.

Last modified: August 10, 2005