Inverse Spectral Problems in Riemannian Geometry

Dartmouth College, Hanover NH

Investigators

Abstract

Abstract Award: DMS-0072534 Principal Investigator: Carolyn S. Gordon Inverse spectral geometry is the study of the extent to which the geometry of a surface or, more generally, of a Riemannian manifold can be extracted from spectral data. The primary spectral data associated to a compact Riemannian manifold are the eigenvalues of the Laplace-Beltrami operator. The investigators propose to apply recently developed methods to study the extent to which the eigenvalue spectrum determines the local geometry of a compact Riemannian manifold. They will also ask the extent to which additional spectral data such as the spectrum of the Laplacian acting on differential forms of various degrees determines the geometry of the manifold. Inverse spectral problems will be considered on Riemannian orbifolds as well as on manifolds; orbifolds are the most tractable singular spaces. For the Schrodinger operator "Laplacian plus potential", the problem of recovering the potential from spectral data will be studied in the case of line bundles over tori. In analogy to the case of planar domains, the lowest eigenvalue of the Laplacian on a compact Riemannian manifold may be viewed as the fundamental tone. The question of whether random Riemann surfaces have large first eigenvalue will be studied using connections between spectra of Riemann surfaces and spectra of graphs. For noncompact Riemannian manifolds, the primary spectral data are the scattering poles; the investigators expect to exhibit continuous families of isopolar metrics. In spectroscopy, one attempts to recover the chemical composition or the shape of an object from the characteristic frequencies of light or sound emitted. In the case of a vibrating membrane such as a drumhead, viewed mathematically as a bounded region in the plane, the spectrum of characteristic frequencies corresponds to the mathematical notion of the Laplace spectrum. The Laplace spectrum is also defined for other geometric objects called manifolds which arise in mathematics and physics. The investigators, along with Scott Wolpert, earlier constructed the first examples of differently shaped drumheads (planar regions) with the same spectrum. Planar regions can differ in their global shape but locally are identical; i.e., if you look at a small piece cut out from one of the regions, you can not tell which region it came from. Recently, the principal investigator developed methods for constructing geometric objects with the same Laplace spectrum but which differ in their local as well as global shape. These methods will be used to investigate which local geometric properties of manifolds are not spectrally determined. Additional spectral problems will also be considered such as the construction of surfaces of arbitrarily large volume but having bounded fundamental tone.