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Publisher:

Chapman & Hall/CRC

Publication Date:

2011

Number of Pages:

422

Format:

Hardcover

Price:

79.95

ISBN:

9781439834596

Category:

Monograph

[Reviewed by , on ]

Michael Berg

09/23/2010

I have had the pleasure of reviewing for MAA Reviews a number of books connected to the Hamiltion Ricci flow and its role in Perelman’s solution of the 3-dimensional Poincaré conjecture: surely a story for the ages, not just because of the scope of Perelman’s achievement, but also because of the controversy surrounding his work when it first appeared on the radar screen and later because of Perelman’s now famous refusal of any and all accolades the mathematical establishment sought to bestow on him, including the Fields Medal and the Clay Prize.

Regarding the early controversy it was the case that, when the mathematical jungle drums first started up, a circle of Chinese mathematicians with Yau at its apparent center placed themselves at odds with Perelman in connection with certain priority matters. These acrimonies were made extra-sticky by Perelman’s work having been launched entirely on the internet, in the form of three rather idiosyncratic articles, often containing sketches rather than full proofs. This ugly business was in due course settled in favor of Perelman (obviously), and his arguments were found to be entirely on the mark. In any event what would otherwise be the story of a heroic quest vouchsafed only to mathematicians and perhaps a tiny number of physicists is now proper fodder for the *New Yorker*. Given Perelman’s insistence on isolation and anonymity there is no small amount of irony in all this.

In any event, in connection with Ricci flow, I have reviewed in this column the books, *Ricci Flow and the Poincaré Conjecture*, by John Morgan and Gang Tian, and *Hamilton’s Ricci Flow*, by Bennett Chow, Peng Lu, and Lei Ni. And now it is my pleasant task to review *Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincaré Conjecture*, by Qi S. Zhang of U. C. Riverside, and I want to start by taking note of a number of synchronicities I find particularly appealing: Zhang was introduced to the subject of Ricci flow by Bennett Chow and Lei Ni in a 2005 workshop on geometric analysis, while it was Gang Tian who recommended the Chinese version of the 2008–2009 Peking / Nanjing lecture notes that evolved into part of the book under review for publication by Science Press Beijing. Additionally, two chapters (the second and the fourth) of *Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincaré Conjecture *spring from a UCR graduate course one of whose note-takers, Jennifer Burke-Loftus, was my student in a number of courses when an undergraduate at my university. It is indeed a small world.

So, Chapters 2 and 4, specifically focused on Sobolev inequalities, are *eo ipso* introductory affairs with respect to the main thrust of the book: these two chapters should be entirely within the province of sufficiently advanced analysis or PDE graduate students. (I always marvel at young scholars like Jennifer who opt to go to the mat with hard analysis, PDE, and so forth: what fortitude!)

Chapter 1 is a well-written and informative introduction; here is a sample: “… a Sobolev inequality states [roughly] that if the derivative of a function is integrable in a certain sense (L^{p}, etc.), then the function itself has better integrability. It lies in the foundation of modern analysis… On the other hand a Sobolev inequality will also yield interesting partial differential equations *via* minimizing the Sobolev constants. It can also reveal useful information on the underlying space or manifold. This last property is the focus of this book.”

Thus, with the game defined early on in this way, Zhang sandwiches in an important (if somewhat austere: have another source handy, maybe) Chapter 3, innocuously titled, “Basics of Riemannian Geometry.” Then, with Chapter 5, gets airborne. To wit, Ricci flow is discussed in the fifth chapter, quickly followed in Chapter 6 by unabashedly *avant garde* stuff: “Perelman’s entropies and Sobolev inequality.” I want to single out §6.4 as particularly piquant: “Harnack’s inequality for the conjugate heat kernel.” This by itself indicates the potential breadth of this material (its depth already being obvious).

Subsequently Chapter 7, “Ancient κ-solutions and singularity analysis,” continues to develop the all-important heat kernel theme, and obviously begins to get down to some of the geometric nitty-gritty. By the way, here’s something relevant from p. 180 of the aforementioned book by Morgan and Tian: “An ‘ancient solution’ is a Ricci flow (M, g(t)) defined for –κ < t ≤ 0 such that for each t, (M, g(t)) is a connected, complete, non-flat Riemannian manifold whose curvature is bounded and non-negative.” Beautiful stuff, no?

And then Zhang’s seventh chapter culminates in § 7.5, “Singularity analysis of 3-dimensional Ricci flow,” introduced as follows: “The main result of this section is the next theorem which says that a space time cube in a 3 dimensional Ricci flow resembles a κ solution provided that the scalar curvature at one point of the cube is sufficiently large.” Very tantalizing. (To find out more about the mysterious κ see p. 209 of the book under review.)

Finally Chapter 8 discusses — briefly — “Sobolev inequality with surgeries” and Chapter 9, the book’s last chapter, is titled, yes, “Applications to the Poincaré conjecture.”

It clear as vodka that, as Zhang advertises in the Preface, “[t]he first half of the book is aimed at graduate students and the second half is intended for researchers.” With some good timing the same reader can start as one and continue as the other. *Sobolev Inequalities, Heat Kernels under Ricci Flow, and the Poincaré Conjecture* looks to be a very important contribution to the genre.

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

**Introduction **

**Sobolev Inequalities in the Euclidean Space **

Weak derivatives and Sobolev space *W ^{k,p}*(

Main imbedding theorem for

Poincaré inequality and log Sobolev inequality

Best constants and extremals of Sobolev inequalities

**Basics of Riemann Geometry**

Riemann manifolds, connections, Riemann metric

Second covariant derivatives, curvatures

Common differential operators on manifolds

Geodesics, exponential maps, injectivity radius etc.

Integration and volume comparison

Conjugate points, cut-locus, and injectivity radius

Bochner–Weitzenbock type formulas

**Sobolev Inequalities on Manifolds **

A basic Sobolev inequality

Sobolev, log Sobolev inequalities, heat kernel

Sobolev inequalities and isoperimetric inequalities

Parabolic Harnack inequality

Maximum principle for parabolic equations

Gradient estimates for the heat equation

**Basics of Ricci Flow**

Local existence, uniqueness and basic identities

Maximum principles under Ricci flow

Qualitative properties of Ricci flow

Solitons, ancient solutions, singularity models

**Perelman’s Entropies and Sobolev Inequality**

Perelman’s entropies and their monotonicity

(Log) Sobolev inequality under Ricci flow

Critical and local Sobolev inequality

Harnack inequality for the conjugate heat equation

Fundamental solutions of heat type equations

**Ancient κ Solutions and Singularity Analysis **

Preliminaries

Heat kernel and

Backward limits of

Qualitative properties of

Singularity analysis of 3-dimensional Ricci flow

**Sobolev Inequality with Surgeries **

A brief description of the surgery process

Sobolev inequality, little loop conjecture, and surgeries

**Applications to the Poincaré Conjecture **

Evolution of regions near surgery caps

Canonical neighborhood property with surgeries

Summary and conclusion

**Bibliography **

**Index**

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