"Advances in Inverse Spectral Geometry"

Outline of Lectures

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I. Introduction.We first introduce the spectrum of the Laplace operator on compact Riemannian manifolds and discuss the variational characterization of the eigenvalues and consequences. We then look at two elementary examples: (i) For flat tori, we see that the spectrum can be explicitly determined and that two flat tori have the same spectrum if and only if they have the same geodesic length spectrum. We then consider a pair of plane domains, each made of 3 congruent rectangles (roughly in a C shape and an S shape) and give Fefferman's elegant proof that they are not isospectral.II. Spectral invariants (What can you hear?)A geometric or topological property is said to be a spectral invariant if it is determined by the spectrum. The primary methods for identifying spectral invariants involve trace formulas, e.g., the asymptotic expansion of the trace of the heat kernel. We discuss the spectral invariants obtained by these methods and describe various applications. In particular, we briefly discuss the use of spectral invariants--especially the ground-breaking work of Osgood, Phillips and Sarnak--in addressing the question: Are isospectral sets of metrics necessarily compact with respect to a natural topology on the space of all metrics?III. Lie group techniques.We now begin to consider methods for constructing isospectral manifolds. We outline the primary methods used to date and then focus on a technique involving representations of Lie groups. As an example, we consider Vignéras' construction of isospectral Riemann surfaces and higher dimensional hyperbolic manifolds.IV. Sunada's Technique.In 1984, T. Sunada gave an elegant and simple technique for constructing pairs of isospectral Riemannian manifolds with a common finite cover. His technique, along with various generalizations, have led to a whole "industry" in constructing isospectral manifolds. In this lecture, we present Sunada's original construction and his heat theoretic proof. We then describe the application of Sunada's technique by P. Buser, R. Brooks and R. Tse to construct isospectral Riemann surfaces of every genus greater than three. Next, a generalization of Sunada's method allows examples presented in Lecture III, as well as other examples constructed by different methods, to be viewed in the context of Sunada's theorem. In fact, we find that all possible isospectral Riemann surfaces arise this way.V. Transplantation of Eigenfunctions; Isospectral Plane Domains.We present another proof of Sunada's theorem, due to P. Bérard and based on ideas of P. Buser, which shows explicitly how to transplant eigenfunctions (standing waves) on one manifold to eigenfunctions (standing waves of the same frequency) on the second manifold. A second geometric transplantation carries closed geodesics on one manifold to closed geodesics of the same length on the second manifold. We apply the method of transplantation to construct pairs of isospectral domains in the plane (work of D. Webb, S. Wolpert and the lecturer). We conclude by listening to D. DeTurck's computer simulation of the sounds produced by these exotic isospectral "drums".VI. Just how strong is Sunada's technique?Sunada's technique and its generalizations account for a large proportion of the known examples of isospectral manifolds. (See Lectures VII-VIII for exceptions.) Is this fact due simply to our lack of knowledge, or do most isospectral manifolds really satisfy the conditions of Sunada's construction? We describe recent work of H. Pesce addressing this question.VII. Isospectral nilmanifolds.Nilmanifolds are compact quotients of nilpotent Lie groups by discrete subgroups. Nilmanifolds have provided a rich source of isospectral manifolds, constructed by a combination of representation theoretic techniques and the use of Riemannian submersions. Examples include, among others, isospectral deformations arising from the representation theoretic techniques of Lecture 3, new examples of isospectral deformations constructed by R. Gornet by different methods, and isospectral nilmanifolds with different local geometry (to be discussed in lecture 8). We will describe the various examples and examine their geometry.VIII. Isospectral manifolds with different local geometry.All examples of isospectral manifolds discussed so far have been locally isometric though they differ in their global geometry. Recently, examples have been constructed which differ in their local geometry as well. These include (i) geodesic balls in different harmonic manifolds (given by Z. Szabo), (ii) closed manifolds with different Ricci curvature, (iii) continuous families of Riemannian metrics on $R^n$ which are isospectral in the sense that there exist unitary operators intertwining the Laplacians, and (iv) continuous families of isospectral compact manifolds with boundary which are not locally isometric. (The latter three constructions are recent work of E. N. Wilson and the lecturer.)IX. Spectral Rigidity.We discuss some spectral rigidity results, focusing primarily on manifolds of negative curvature and on convex planar domains.X. Open questions and concluding remarks.

In their foundational paper, Gordon and Wilson constructed isospectral deformations on 2-step nilpotent manifolds. At approximately the same time, Sunada gave a finite-groups approach to constructing isospectral manifolds. These two results were combined by DeTurck and Gordon to give a Sunada-type approach to isospectral manifolds. We will discuss another way to combine these results, to give a construction of isospectral surfaces which may be thought of as discretizing the nilpotent deformation approach. The result is the construction of large numbers of mutually isospectral Riemann surfaces. (This is joint work with R. Gornet & B. Gustafson.)

Let (M,g) be a closed Riemannian three-manifold. One measure of the geometric content of the spectrum is theisospectral set, i.e., the set of all closed Riemannian manifolds (M',g') with the same Laplace spectrum as (M,g). We review recent work of Brooks, Perry, and Petersen showing that for a spectrally determined neighborhood of certain metrics g, the isospectral set is compact in a sense we make precise.

(Joint work with Pierre Bérard.) We exhibit a pair of flat bordered surfaces which are isospectral for the Neumann boundary conditions, one of which is orientable while the other is nonorientable. The surfaces are constructed using the orbifold version of Sunada's theorem, and Neumann isospectrality is verified explicitly by transplantation of eigenfunctions. By using a fundamental tile which is as symmetrical as possible, we show that our construction yields a pair of Neumann isospectral bordered surfaces which are not Dirichlet isospectral.

For a manifold a sequence of eigenfunctions gives rise to a geodesic flow invariant measure on the unit tangent bundle. A simple construction of such measures for hyperbolic surfaces will be presented. It is an open problem to characterize all possible limiting measures. For PSL(2;Z) a result of Luo-Sarnak about such limits will be recast to give an asymptotic formula combining the divisor function and the Riemann zeta function.

The trace of the wave group $e^{it \sqrt{\Delta}}$ on a compact Riemannian manifold (M,g) with simple length spectrum gives rise to a sequence of spectral invariants $a_{\gamma k}$ associated to each closed geodesic $\gamma$. We will characterize the wave invariants in the same sense that the heat invariants are characterized as integrals of homogeneous curvature polynomials, explain how they can be calculated and discuss some possible applications.