Inverse Spectral Geometry

In this paper, we would like to sketch a picture aimed at giving a comprehensive answer to the question: how does one go about reconstructing a manifold M from the spectrum of its Laplace operator ? It is understood that, in general, there is no unique way of reconstructing M, because a manifold is not in general uniquely determined from its spectrum. So let us make the following deenition: Deenition 0.1 A manifold M is compactly determined by a set of conditions P (which M satisses) if there is a nite set fM 1 compact in the C 1 topology, such that any manifold M 0 which satisses P is isometric to a manifold lying in one of the M i 's. We then have the following: Conjecture 0.1 Every compact manifold is compactly determined by its spectrum. We are still fairly far away from this conjecture in its full generality, although we remark that we can obtain the conjecture if we add to the spectrum some curvature assumptions which in other areas of geometry would be regarded as fairly weak. Our focus in this paper will be on a presentation of techniques which, when used in combination, allow one to attack the main conjecture. Our feeling is that the main technical components necessary to establish the conjecture are in fairly good shape, and we would be surprised if a radically diierent approach would be required, or even helpful, to arrive at the nal destination. With that said, however, in each section there are topics and problems which remain unexplored, and whose solution would be a major step towards the solution of the main conjecture. It is our pleasure to set out here our view of what these problems are. Our emphasis here will be on setting out how various techniques are used, rather than how they are proved, although we have not shied away from sketching a proof when we thought it would be illuminating. The plan of the paper is as follows: in x1, we give an overview of ideas related to the Cheeger Finiteness Theorem Ch2] and its geometric relatives. This is the main technique by which one builds up a \rough model" of a man-ifold from geometric data. In x2, we discuss bootstrapping techniques. These 1 techniques serve two important purposes: rstly, they allow one to \smooth out" the rough models of x1. Secondly, by examining …