Inverse Spectral Problems in Riemannian Geometry

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AbstractAward: DMS-0072534Principal Investigator: Carolyn S. GordonInverse spectral geometry is the study of the extent to which thegeometry of a surface or, more generally, of a Riemannianmanifold can be extracted from spectral data. The primaryspectral data associated to a compact Riemannian manifold are theeigenvalues of the Laplace-Beltrami operator. The investigatorspropose to apply recently developed methods to study the extentto which the eigenvalue spectrum determines the local geometry ofa compact Riemannian manifold. They will also ask the extent towhich additional spectral data such as the spectrum of theLaplacian acting on differential forms of various degreesdetermines the geometry of the manifold. Inverse spectralproblems will be considered on Riemannian orbifolds as well as onmanifolds; orbifolds are the most tractable singular spaces. Forthe Schrodinger operator "Laplacian plus potential", the problemof recovering the potential from spectral data will be studied inthe case of line bundles over tori. In analogy to the case ofplanar domains, the lowest eigenvalue of the Laplacian on acompact Riemannian manifold may be viewed as the fundamentaltone. The question of whether random Riemann surfaces have largefirst eigenvalue will be studied using connections betweenspectra of Riemann surfaces and spectra of graphs. Fornoncompact Riemannian manifolds, the primary spectral data arethe scattering poles; the investigators expect to exhibitcontinuous families of isopolar metrics.In spectroscopy, one attempts to recover the chemical compositionor the shape of an object from the characteristic frequencies oflight or sound emitted. In the case of a vibrating membrane suchas a drumhead, viewed mathematically as a bounded region in theplane, the spectrum of characteristic frequencies corresponds tothe mathematical notion of the Laplace spectrum. The Laplacespectrum is also defined for other geometric objects calledmanifolds which arise in mathematics and physics. Theinvestigators, along with Scott Wolpert, earlier constructed thefirst examples of differently shaped drumheads (planar regions)with the same spectrum. Planar regions can differ in theirglobal shape but locally are identical; i.e., if you look at asmall piece cut out from one of the regions, you can not tellwhich region it came from. Recently, the principal investigatordeveloped methods for constructing geometric objects with thesame Laplace spectrum but which differ in their local as well asglobal shape. These methods will be used to investigate whichlocal geometric properties of manifolds are not spectrallydetermined. Additional spectral problems will also be consideredsuch as the construction of surfaces of arbitrarily large volumebut having bounded fundamental tone.