Spectral Geometry of the Laplace-Beltrami Operator

Abstract

This thesis aims to study certain geometric properties of the eigenvalues of the Laplace-Beltrami operator in the general setting of Riemannian manifolds. Starting with a detailed study of prerequisites like Riemannian geometry and spectral theory for Laplacian, the primary focus is on the upper bounds for the closed eigenvalues in the conformal class of a compact Riemannian manifold (M, g). We look at the interplay of geometric quantities like curvature, min-conformal volume, etc., with the min-max variational characterization of eigenvalues (using Rayleigh quotient) in obtaining the desired upper bounds that are asymptotically consistent with the Weyl law. We study certain powerful techniques from metric geometry that are not only the key ingredients in proving the main results, but also have far reaching applications in many other contexts. Independent of this, the section on inverse spectral geometry focuses on a historic counterexample that negatively answers Mark Kac’s famous question, “Can one hear the shape of a drum?”