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Ricci flow and the Poincaré conjecture. (English) Zbl 1179.57045

Clay Mathematics Monographs 3. Providence, RI: American Mathematical Society (AMS); Cambridge, MA: Clay Mathematics Institute (ISBN 978-0-8218-4328-4/hbk). xlii, 521 p. (2007).
A Ricci flow on a smooth manifold \(M\) is a one-parameter family of metrics \(g(t)\) on \(M\) (where \(t\) is in some interval) satisfying the evolution equation
\[ \frac{\partial g(t)}{\partial t} = -2\operatorname{Ric}(g(t)), \]
in the space of metrics. Richard Hamilton first considered this flow over a quarter century ago and proved the fundamental result that if a manifold starts with a metric of positive Ricci curvature, then the Ricci flow “shrinks” the manifold to a point in finite time (with a consequent blowing up of curvature). Of course, by this we mean that the metric evolves so as to have a finite time singularity. The sphere has exactly this behavior and, of course, possesses a constant sectional curvature metric, so we have a hint as to the nature of the Ricci flow.
It is perhaps easier to understand an equivalent flow called the normalized Ricci flow. Suffice it to say that this normalized flow preserves volumes, so cannot shrink manifolds to points. Furthermore, the special metrics that are rest points under the flow are the Einstein metrics: that is, those metrics whose Ricci curvatures are constant multiples of the metric. It is an amazing dimensional consequence that all Einstein metrics on 3-manifolds are constant sectional curvature metrics. In this normalized framework, Hamilton’s theorem says that the Ricci flow goes to an Einstein metric – hence a metric of constant positive curvature. Of course, if the manifold \(M\) is simply connected, then \(M\) is a sphere. When \(M\) does not start with positive Ricci curvature, then the flow can have finite time singularities. The idea then (due to Hamilton also) is to do surgery on a neighborhood of the singularity (thus changing the topology of the manifold) and continue the Ricci flow to the next singularity etc. The goal would then be to decompose the manifold as a connected sum using a finite number of surgeries – but this is not automatic.
G. Perelman’s great step forward was to show that if the prime decomposition of the (orientable, say) 3-manifold \(M\) does not have aspherical prime summands, then the Ricci flow exists for all time and becomes extinct after a finite time (i.e. uses up the whole manifold in finite time and thus gives a connected sum decomposition of \(M\)). (The reader should note that some confusion might take place since the book under review uses the term “acyclic” instead of the correct “aspherical”.) Furthermore, this implies that \(M\) is diffeomorphic to a connected sum of \(3\)-dimensional spherical space forms and \(S^1 \times S^2\)’s (in the orientable case). Of course, if \(M\) is simply connected, none of the latter arise and the only space form allowed is the (simply connected!) sphere itself; hence, \(M\) is the sphere – and this is the Poincaré conjecture; any simply connected closed \(3\)-manifold is diffeomorphic to \(S^3\). The technical details of Perelman’s work are forbidding. For instance, he must classify the neighborhoods of singularities and this involves notions of convergence of Riemannian manifolds (i.e. Cheeger-Gromov convergence) and a fundamental non-collapsing result. The identification of these canonical neighborhoods allows surgery to be done in a way so that we still understand the topology of the new manifold and so that Ricci flow can be continued after the surgery.
The present book is a tour-de-force of geometro-analytic technique that fills in all details of Perelman’s proof of the Poincaré conjecture (or a bit more in fact). The casual reader might just want to peruse the Introduction where all the basic ideas are discussed. The first five chapters gives background in differential geometry and Ricci flow, although it must be said that a reader coming to this material for the first time will have a very difficult time. Indeed, these background chapters are more refreshers for readers with some experience in the areas. For nice introductions to the Ricci flow, see [B. Chow and D. Knopf, The Ricci flow: an introduction. Mathematical Surveys and Monographs 110, Providence, RI: American Mathematical Society (AMS) (2004; Zbl 1086.53085) and B. Chow, P. Lu and L. Ni, Hamilton’s Ricci flow. Graduate Studies in Mathematics, 77. Providence, RI: American Mathematical Society (2006; Zbl 1118.53001)]. Chapters 6 through 12 deal with Perelman’s innovations: in particular, his length function, general non-collapsing results and discussion of geometric limits. Part 3 (chapters 13 through 17) focuses on the existence of Ricci flow with surgery. Finally, part 4 (chapters 18–19) completes the proof of the Poincaré conjecture. For any topologist or geometer who actually wants to understand the proof of the Poincaré conjecture, this book is a godsend.

MSC:

57R60 Homotopy spheres, Poincaré conjecture
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
35K55 Nonlinear parabolic equations
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
57M40 Characterizations of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010)
57N12 Topology of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010)

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