Hamilton's Ricci Flow

The Poincaré Conjecture, that every closed, compact 3-manifold which is connected and simply connected is diffeomorphic to the 3-sphere, is without doubt the hottest mathematical topic around these days, both because of Grigory (“Grisha”) Perelman’s huge achievement in proving it and because of his subsequent refusal of the Fields Medal. (Apparently he’s not too keen on the Clay Institute’s million dollars either, if and when it’s offered.)

We all know, too, as if this wasn’t enough, that a major controversy surrounds Perelman’s solution insofar as a small circle of mathematicians centered around S. T. Yau have raised questions about priority vis á vis the critical detailed application of Richard Hamilton’s now famous Ricci flow to the geometrization conjecture of Bill Thurston, known to imply Poincaré (which is how Perelman did it). These de facto competitors to Perelman, X. P. Zhu and H. D. Cao primary among them, published detailed arguments in lieu of Perelman’s sketches (for lack of a better word) and soon the question arose of who should get credit for what — and when? Evidently the expert judges’ verdict is that, to be sure, Perelman’s sketches disclose that he was indeed in possession of and in a position to give a full and complete proof.

Given the cross-over appeal of this tale (how could any media player pass up the chance of once again highlighting the foibles and eccentricities of mathematicians, now that the public fascination with Andrew Wiles has waned?), many popular articles have appeared on the topic of Perelman vs. Yau et al., e.g. “Manifold Destiny,” a recent New Yorker offering by Sylvia Nasar (of A Beautiful Mind fame) and David Gruber. Apparently this was the straw that broke the camel’s back as far as Yau’s camp was concerned; Yau mobilized his attorney to attack The New Yorker claiming that Nasar and Gruber’s portrayal of Yau and his champions was defamatory. Happily, however, more recently peace seems to be breaking out all over as is evidenced by the following passage from Yau’s April 2007 Letter to the Editor of the Notices of the AMS: “The achievements of Hamilton and Perelman in solving the conjecture… are unparalleled. They far exceed the established standards for Fields Medals. I fully support, and have always said so, the award of the Fields Medal to Perelman…”

Yau goes on to say that “Hamilton clearly deserves the Fields Medal also, but he is not eligible at this time because of age restriction.” This said, and acknowledged, it is perhaps particularly apt that the book that forms the subject of the present review should be titled Hamilton’s Ricci Flow. The authors are Bennett Chow and Lei Ni of the University of California at San Diego and Peng Lu of the University of Oregon; Chow is also affiliated with East China National University. The UCSD connection is particularly noteworthy, seeing that not only were Hamilton and Yau on the faculty there some twenty years ago, but so was Michael Freedman, who had just received his Fields Medal for proving the 4-dimensional Poincaré Conjecture. With this kind of geographical connection it’s not surprising that Chow, Lu, and Ni (or CLN for short) seek to underscore Hamilton’s work, given what a lion’s share of the credit should obviously go to him.

CLN refer to Hamilton’s Ricci Flow as Γ_ijk (“the connection”) in contradistinction to the 2004 AMS publication, The Ricci Flow: An Introduction, by Chow and Dan Knopf, which they refer to as g_ij (“the metric”). Chow and others have also started a new series of books with AMS, The Ricci Flow: Techniques and Applications.

We encounter the following rather enigmatic phrase in the Preface to the present book: “We would like to imagine g_ij as an attempt to write in the style of jazz and Γ_ijkas an attempt to write in the style of rock ‘n’ roll.” I must say that as a frustrated jazz guitarist, who played some rock ‘n’ roll in graduate school (about twenty years back — at UCSD!), this is a remark I simply can’t leave alone. What do CLN mean? Does it really apply? Well, here is what they say:

In g_ij we dive right into the Ricci flow [a nice bit of imagery…] and then proceed at a metric pace, taking the time to appreciate the intricacies and nuances of the melody and structure of the mathematical music. In Γ_ijk, after starting from more basic material, as a connection to the Ricci flow, the tempo is slightly more upbeat. The recital is defined on a longer page interval, and consequently more ground is covered, with the intention of leading up to the forefront of mathematical research. In g_ij calculations are carried out in detail in the main body of the book whereas in Γ_ijkthe details appear either in the main body or at the end of the book in the exercises. [Caveat!] With the addition of basic material on Riemannian geometry and a substantial number of solved exercises, Γ_ijkis accessible to graduate students and suitable for use in a semester or year-long graduate course.

CLN go on to provide two useful Leitfadenen, which is to say, an over-all structure of the book and a two-semester course out-line.

This is all good and well, but it begs the question of whether one should learn to play jazz before rock ‘n’ roll (as I did), or the other way around. Though I think this question ultimately only has a subjective answer, it’s largely the case that rock ‘n’ roll is easier to play than jazz, so going at Γ_ijk first is probably a good idea. However, I have of late developed a research-need to learn more about these aspects of geometry and topology (and PDEs: the Ricci flow is given by ∂ g_ij/∂ t = -2R_ij), so I’m going to get a copy of g_ij from the AMS in the near future so that I might be able to learn to play a little in both styles. In any case, there is no doubt that CLN mean Γ_ijk as an advanced graduate text.

The corresponding graduate course would be pretty dense. Γ_ijk consists of some 600 pages of serious — and relatively young — mathematics, and manifestly requires a sound preparation on the part of the student or the reader, even if the book’s first chapter, “Riemannian geometry,” seeks in the first 100 pages to provide the geometrical foundation for what follows in the next 500 pages. This is pretty austere stuff: already on p. 3 we read “Recall [!] that the Levi-Civita connection (or Riemannian covariant derivative)… is the unique connection on [the tangent bundle] that is compatible with the metric and is torsion-free.” This is a bit much for an outsider like me, so my feeling is that one shouldn’t crack the pages of this book, or certainly not the last 500 pages, without mastering the equivalent of what’s in Chapter 1 at the level of a pretty advanced differential geometry student, with his first course (or two) already under his belt.

What about the latter 500 pages, then? Well, they are arranged into eleven chapters ranging from “Fundamentals of the Ricci Flow equation” to “Space-time geometry” (Einstein redux) with coverage in between of closed 3-manifolds with positive curvature, Perelman’s famous “no local collapsing,” singularity analysis, and Harnack estimates. This weighty material is completed by two appendices, “Geometric analysis related to the Ricci flow” and “Analytic techniques for geometric flows.” Interestingly, these two sections of the book, each at between 30 and 40 pages, include some generally familiar material, e.g. the heat kernel and the heat equation, Green’s function, eigenvalues of the Laplacian, and Riemann surfaces per sé. Still, it’s all enlisted in the cause of explicating the Ricci flow (to wit, on p. xiii we encounter the pithy sentence: “The Ricci flow is like a heat equation for metrics”). Finally, there are some 40 pages of solutions to exercises.

Γ_ijk, which is to say, Hamilton’s Ricci Flow, is a thorough coverage of a hugely important subject at the frontier of current research, possessed of connections with sundry other parts of mathematics, including in particular general relativity. It will certainly take an advanced graduate student to the threshold of research proper. I do suggest that the prospective reader of this book prepare himself properly for what lies ahead and keep a few supplementary sources (especially on Riemannian geometry) close at hand, however.

Michael Berg is Professor of Mathematics at Loyola Marymount University in California.