The author of this book notes that his focus falls on the “linear theory” aspects of geometric analysis, specifically the question of the eigenvalues of the Laplacian and the connection this theme has with, in particular, “the study of non-linear partial differential equations that arise in geometry.” This means, of course, that the prevailing setting, at least as far as the geometry is concerned, is that of Riemannian manifolds; the book accordingly starts out with a discussion of the important question of “[the f]irst and second variational formulas for area.” This approach presupposes some familiarity on the reader’s part with Riemannian geometry *per se*, and this suggests as an implicit prerequisite a decent facility with the material covered in, for instance, Steven Rosenberg’s excellent *The Laplacian on a Riemannian Manifold*, and/or Isaac Chavel’s (also excellent, but very differently flavored) *Riemannian Geometry: A Modern Introduction*. Rosenberg’s text specifically advertises itself as “an introduction to analysis on manifolds,” and therefore probably has the edge as far as forming a proper prelude to the present book is concerned: Peter Li’s objective is expressly analytic, after all. But that’s not all…

In short order Li gets to volume comparison and Laplacian comparison theorems, and by around page 40 he has hit the topic of Poincaré’s inequality and the first eigenvalue (of the Laplacian), or more specifically the smallest non-zero eigenvalue of the Dirichlet Laplacian. One of the salient points is that by a 1976 theorem of Cheng, this value is dominated by the first Dirichlet eigenvalue of a certain associated geodesic ball. The proof of this result is already sporty indeed, combining hard analysis (maximum principles, Lipschitz functions, Dirichlet boundary conditions, &c.), with some pretty sophisticated geometry and such business as Rayleigh’s principle for eigenvalues. This gives the flavor of what we’re dealing with here: the material on Poincaré’s inequality and the first eigenvalue mentioned above culminates some 10 pages later in a 1980 theorem of the author and S-T. Yau bounding this eigenvalue below by a value depending on the diameter of the underlying manifold (taken to be compact and of course finite dimensional) and its Ricci curvature. Very deep and serious stuff.

And it goes on from there. Sobolev space methods come in, in spades, the heat equation is (naturally) prominently featured, and the intensity rises and rises. This is pretty austere mathematics, and certainly is not for rookies.

Geometric analysis is one of the hottest areas in mathematics these days, as illustrated for instance by its appearance in Perelman’s proof of the Poincaré Conjecture. But Peter Li’s *Geometric Analysis* is not focused on such topological questions: it is analysis, unapologetically and with extreme prejudice. Every page bespeaks this message. Indeed, the book is meant to lead properly prepared researchers in this area more deeply into the field, as is borne out by the fact that much of the included material was originally presented in various research workshops and advanced courses, from MSRI to Seoul and from UC Irvine to the fourteenth “Escola de Geometria Differencial” in Brazil. Thus, for the right audience of analysts (and, I suppose, differential geometers of a certain disposition), this book is indeed right on target. But, again, it is by no means for the non-initiated.

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