If g is a geodesic on a compact Riemann surface of genus at least 2, write N(g) for its norm, the exponential of its length. Every closed geodesic can be expressed as a power of a prime geodesic. Now, if, with x a positive real number, π(x) denotes the number of prime geodesics of norm at most x, then, as x tends to infinity, π(x) is asymptotic to … x/log(x), just as in the Prime Number Theorem.
This truly wonderful fact is covered in Chapter 9, §4 of the book under review, which the author initiates with the comment that “[a]s Selberg points out … there is a strong analogy between the prime numbers and the elements in a discrete subgroup of SL(2,R), or, in our case, the lengths of the closed geodesics.” So, the cat at least begins to peek out of the bag: once discrete subgroups of SL(2,R) are mentioned, every number theorist’s antennae start to quiver and even the remarkable theorem phrased above soon takes on something of a predictable quality — these beautiful parallels are quite exact, after all.
But there is nonetheless something very deep at work here and not only §4 but the remainder of Chapter 9 is devoted to disclosing much of this subtle connection: Selberg’s trace formula is discussed in §5 and §6 includes a discussion of contemporary developments regarding the error terms in the Prime Number Theorem for compact Riemann surfaces of genus ≥ 2 building on work by Huber; beyond this §7 concerns hyperbolic lattice points. Quite a wealth of “hyperbolic” connections to mainstays of classical analytic number theory are dealt with in the middle of this remarkable book.
Only a number theorist (like me) would cast Peter Buser’s Geometry and Spectra of Compact Riemann Surfaces in a supporting role vis à vis number theoretic themes. In point of fact the book is, by the author’s own description, devoted to “two subjects [:] … the geometric theory of compact Riemann surfaces of genus greater than one [i.e. ≥ 2] … [and] the Laplace operator and its relationship with compact Riemann surfaces.”
The latter topic is of huge importance in modern mathematics, evincing connections to such marvelous themes as the heat equation, Hodge theory, and the Atiyah-Singer index theorem. (A wonderful book in this regard is S. Rosenberg’s The Laplacian on a Riemannian Manifold.) Buser’s coverage of this material is very impressive indeed, with his Chapter 7 devoted to the ever-so-important spectrum of the Laplacian, and with his book closing with Chapter 14, “Perturbations of the Laplacian in Hilbert space,” a particularly dense and rich chapter.
These heavier themes are preceded, in the fist half dozen chapters of the book, by what Buser presents as “an introduction to the geometry of compact Riemann surfaces based on hyperbolic geometry and on cutting and pasting.” Buser goes on to say that “[t]his part is in textbook form at about graduate level … [with] prerequisites … kept to a minimum, but I assume that the reader has a background in differential geometry or in complex Riemann surface theory.”
The first part of Geometry and Spectra of Compact Riemann Surfaces is a pleasure to read. There is a lot of motivation given, examples proliferate, propositions and theorems come equipped with clear proofs, and excellent drawings (or computer renderings, I guess) abound.
Then, as already discussed above, the second part of the book is focused on the Laplacian: we read on the back-cover of the book that this “second part … is a self-contained introduction to the spectrum of the Laplacian based on the heat equation,” and Buser covers a lot of ground indeed. In addition to what he does in Chapter 9, on, so to speak, hyperbolic analytic number theory, he addresses such topics as small eigenvalues and the minimax principles, Wolpert’s theorem (“a generic compact Riemann surface is determined up to isometry by its length or eigenvalue spectrum”), and Sunada’s theorem (“the covering technique which leads from Grassmann equivalent groups to non-solitary number fields works also in Riemannian geometry and leads to isospectral manifolds”).
Finally, I must pick at a nit, which is really quite dissonant in connection with a book that is this well-written: Chapter 14 is listed in the table of contents as “Perturbations of the Laplacian in Hilbert space,” while on p. 362, the chapter’s first page, we read “Perturbations of the Laplacian in Teichmüller space.” As this important chapter’s first sentence reads, “An important consequence of the real analytic structure of Teichmüller space is the analyticity of the Laplacian,” it looks like the latter choice is the right one. By the way, this chapter contains something useful in and of itself: “All analyticity arguments are reduced to a few elementary properties of holomorphic functions of several complex variables such as the Cauchy integral formula and Weierstrass’ convergence theorem” — a minicourse in its own right.
Geometry and Spectra of Compact Riemann Surfaces is a fine piece of scholarship and a pedagogical treat.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.