Inverse spectral problems on compact Riemannian manifolds

Abstract

These notes are a revised and expanded version of lectures given at the Nordic Summer School of Mathematics in August of 1988. Their purpose is to publicize in the Mathematical Physics community some of the very interesting inverse spectral problems of current interest in differential geometry. These questions concern the geometric content of the spectrum of the Laplacian. To what extent does the Laplace spectrum of a Riemannian manifold determine its metric? Numerous counterexamples show that the spectrum of the Laplacian does not uniquely determine a Riemannian metric on a fixed smooth manifold M. One measure of the nonuniqueness is the "size" of the isospectral set of a given Riemannian metric g on M, that is, the set of all non-isometric Riemannian metrics g~ on M with the same Laplace spectrum as g. These lectures motivate and describe recent work of Osgood, Phillips and Sarnak [38, 39, 40], Brooks, Perry and Yang [6] and Chang and Yang [9, 10] which prove compactness theorems about the isospectral set of a Riemannian metric in various situations. Along the way we give some of the history of inverse spectral problems on open domains in R ~ and compact manifolds. These lectures cover only a very narrow area of research activity in a vast field: interested readers should consult the books of Chavel [12] and Berger, Gauduchon, and Mazet [4] to obtain a broader perspective. Another survey of the results we discuss, written from a different point of view, has recently been given by Chang and Yang [11]. For background material on Riemannian geometry, the Laplacian, and the spectrum of the Laplacian, readers can consult [4, 12] and [14], §11.3, 12.1, 12.5, and 12.6. My own involvement in the field has begun quite recently, in pleasant collaboration with Robert Brooks and Paul Yang. I am grateful to them for the insights that I have gained and for those that they have shared with me. I am also grateful to Peter Sarnak for helpful discussions, and to Stanford University for hospitality during part of the time that these lectures were written. During that time, Alice Chang and Paul Yang were kind enough to give m e private lectures on their work, some of which is described below. I am also indebted to Rare Mazzeo, Brad Osgood, and Charles Yeomans for their careful reading of these notes and their helpful comments. Finally, I would like to thank the organizers and students of the Nordic Summer School of Mathematics for the opportunity to give these lectures, and for the attentiveness with which they were received!