Spectral Theory in Riemannian Geometry

The title by itself is tantalizing: this is very exciting mathematics on any number of counts. For example, spectral theory is arguably the centerpiece of functional analysis and famously interacts with quantum mechanics; on the other hand, Riemannian geometry is instrumental in such marvels as general relativity, any number of aspects of Morse theory in its broadest sense (e.g. the work of Novikov), and Perelman’s recent proof of the 3-dimensional Poincaré conjecture. So it is, then, that the material covered in the pages of this book attests to the fecund interplay between mathematics and physics over the centuries, and its recent waxing.

We start off with what Lableé describes as a detailed introduction to spectral theory, with the first subsection titled, “From P.-S. Laplace to E. Beltrami,” and we close the book, about 170 pages later, with a discussion of Thurston’s Geometrization Conjecture, the aforementioned Poincaré Conjecture, and Perelman and the Ricci flow. In between we meet Laplacians, heat kernels, the Schrödinger equation, Hodge-de Rham theory, harmonic oscillators, large eigenvalues à la Weyl, and even the following charming variant on Kac’s famous question: “Can one hear the holes of a drum?” (Find out the answer in Chapter 6). Again, all very tantalizing.

The book is arranged in a very appealing and effective fashion: the earlier detailed introduction to spectral theory (including a discussion of unbounded operators on Hilbert space) is followed by a thorough treatment of the Laplacian on a compact Riemannian manifold, after which the Laplacian’s spectrum is analyzed. This material culminates in Chapter 4 with the minimax theorem, followed by a discussion of the Schrödinger operator and the Hodge-de Rham Laplacian. With Schrödinger first appearing in §4.1 and then headlining in §4.6, the chapter’s closing section, it is most welcome to see such an emphasis on the Schrödinger picture of quantum mechanics. After all it provides one of the single most spectacular illustrations of the efficacy of spectral theory in other parts of mathematics (and, yes, physicists notwithstanding, their subject is a proper subset of ours).

After this we get to “direct problems in spectral geometry,” where the dominant geometry is nonetheless off set a bit by the inclusion of a brief discussion of the harmonic oscillator. The aforementioned discussion of the question of whether one can “hear the holes of a drum,” is given as an intermezzo, and the book closes with a very “hot” discussion of “inverse problems in spectral geometry.” Here Kac is given his due (actually, according to Wikipedia, the phrasing of the question is due to Lipman Bers, the problem itself going back to none other than Hermann Weyl): this is of course a classic example of an inverse problem. And, indeed, it soon gets hotter and hotter: the heat kernel is treated in §7.5 (with the heat equation appearing already in §4.1, just before Schrödinger’s entrance — a nice juxtaposition). The book closes with a “[f]ew words about Laplacian and conformal geometry,” where we encounter spectral zeta functions (warming my number theorist’s heart), the Ricci flow, and 3-manifolds: hot stuff indeed.

I really like this book and its strictly-business style. It contains good problem sets (which the serious reader should do, of course), and reads well. I learned what I know about this subject from the excellent but very compact book, The Laplacian on a Riemannian Manifold, by Steven Rosenberg. With its broad scope the book under review should make a wonderful supplement, or, given that Lablée intends it as an introduction to the subject, a prequel, as they say in the movies. 

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.