A fundamental problem in differential geometry is to find canonical metrics on Riemannian manifolds, i.e., metrics which are highly symmetrical, for example metrics with constant curvature in some sense. In 1982, R. S. Hamilton [J. Differ. Geom. 17, 255–306 (1982; Zbl 0504.53034)] introduced the concept of Ricci flow, that is a method of processing a smooth Riemannian metric \(g\) on a fixed, smooth, closed manifold \(M\) by allowing it to evolve as a one-parametric family of Riemannian metrics \(g_t\), where \(t\) is in some interval, under the weakly parabolic differential equation \(\frac{\partial g_t}{\partial t}=-2 \operatorname{Ric}(g_t)\), with \(g_0=g\), where Ric\((g_t)\) is the Ricci tensor field associated to \(g_t\). The Ricci flow is a natural analogue of the heat equation for Riemannian metrics, and consequently, the curvature tensor fields evolve by a system of diffusion equations which tends to distribute the curvature uniformly over the manifold, so that one expects that the initial metric should be improved and evolve into a canonical metric. This concept was the essential ingredient of Hamilton’s method, allowing him to prove the theorem asserting the existence of a Riemannian metric of constant positive sectional curvature on a closed three-manifold \(M\), if the initial metric \(g\) has the property \(\operatorname{Ric}(g)>0\). More precisely, under these assumptions about \((M,g)\), Hamilton (loc. cit.) proved that the Ricci flow has a finite-time singularity and, at the singularity, the diameter of \(M\) tends to zero and the curvature blows up at every point. Also, he showed that – in this case – rescaling by a time-dependent function, so that the diameter is constant, produces a one-parameter family of Riemannian metrics, converging smoothly to a Riemannian metric of constant positive sectional curvature.
The theory of the Ricci flow was extensively studied by R. S. Hamilton and many others [for a collection of papers making readily available in one book the works of Hamilton and others on Ricci flow see H. D. Cao (ed.) et al., ‘Collected Papers on Ricci Flow’, Somerville, MA: International Press (2003; Zbl 1108.53002)] and outstanding achievements had happened during the last years via this theory. In fact, R. S. Hamilton was the originator of a well-known program, developed together with S.-T. Yau, for understanding the topology of manifolds and for working out the analysis related to the Ricci flow in order to approach the full Thurston Geometrization Conjecture [W. Thurston, Bull. Am. Math. Soc., New Ser. 6, 357–379 (1982; Zbl 0496.57005)], which provides a way to classify all three-manifolds (see Hamilton’s talk at the 2006 ICM in Madrid). The toughest obstacle within this program was handling the singularities that could develop in the Ricci flow. The deep new ideas in geometric analysis introduced by G. Perelman allowed him to show that in three dimensions, there is a well-defined Ricci flow with surgery procedure, which is the Ricci flow punctuated at a discrete set of times by carefully chosen surgery operations, which makes it possible to change the topology of the manifold with a remarkable property [see his three papers posted on the arXiv:org/math.DG/0211159v1 (2002), 0303109v1 (2003), 0307245v1 (2003); see also the expository monographs of B. Kleiner - J. Lott and H.-P. Cao - X.-P. Zhu, quoted below, for a detailed account on Perelman’s arguments, including several clarifications and alternative derivations on certain difficult parts of his papers]. These made possible, after large collaborative projects [see, for details, A. Jackson, Notices Am. Math. Soc. 53, No. 8, 897–901(2006)], to complete Hamilton’s program, and – in particular – to settle two major conjectures in geometry and topology, namely the previous mentioned Thurston’s conjecture and Poincaré’s conjecture [see H.-D. Cao and X.-P. Zhu, Asian J. Math 10, 145–492 (2006; Zbl 1200.53057); B. Kleiner and J. Lott, “Notes on Perelman’s papers”, arXiv:math.DG/0605667, 25 May 2006; the forthcoming volume “Structure of Three-Dimensional Space: The Poincaré and Geometrization Conjectures”, International Press of Boston; J. W. Morgan and G. Tian, “Ricci Flow and the Poincaré Conjecture”, a book which is to appear in early 2007 and now posted on the arXiv (July 27, 2006), a 473-page manuscript; T. Tao, “Perelman’s proof of the Poincaré conjecture: A nonlinear PDE perspective”, arXiv:math.DG/0610903v1, 29 October 2006]. Moreover, Perelman’s work suggests the way for further progress in these fields.
This book is a gentle, concise introduction to the subject in the light of Perelman’s work from 2002/2003 (loc. cit.) [for other very recent texts that are self-contained, modern introductions to the Ricci flow, covering also other basic important topics, published after the author’s book, see R. Müller, “Differential Harnack Inequalities and the Ricci Flow”, EMS Series of Lectures in Mathematics. (Zürich): European Mathematical Society Publishing House. (2006; Zbl 1103.58014); B. Chow, P. Lu, and L. Ni, “Hamilton’s Ricci Flow”, Graduate Studies in Mathematics, Vol. 77 (2006) (expected publication date is January 18, 2007)].
Chapter 1 explains the concept of Ricci flow and where it came from, it presents explicit simple examples of such flows, it gives special solutions (in particular, Ricci solitons) of the flow equation, it lists all closed, oriented manifolds in low dimensions and sketches what is known about the geometric structures they support, and it gives an heuristic outline concerning the use of Ricci flow to prove topological and geometric results.
Chapter 2 is on Riemannian geometry and covers Einstein metrics, deformation of geometric quantities as the Riemannian metric varies, Laplacian of the Riemann tensor field \(Rm\) as well as the evolution of \(Rm\), the Ricci tensor field Ric and the scalar curvature \(R\) under the Ricci flow.
Chapter 3 discusses consequences of the maximum principle. By applying this principle, the author illustrates the possibility to get some preliminary control on how \(R\), the volume and the norm \(| Rm| \) of \(Rm\) evolve as well as a global version of estimates for higher order derivatives of \(\nabla^kRm\) (=the \(k\)-th order covariant differential of \(Rm\) of a Ricci flow).
Chapter 4 sketches the type of existence and uniqueness results one can prove for reasonable parabolic PDE.
Chapter 5 justifies the existence and uniqueness of a Ricci flow, over some short time interval, starting with a given smooth initial metric, on a smooth, closed manifold. This is done by reducing the problem to the solution of a parabolic equation using the D. DeTurck trick [J. Differ. Geom. 18, 157–162 (1983; Zbl 0517.53044)]. Moreover, taking into account estimates established in Chapter 3, the fact that Riemann tensor field \(Rm\) of a Ricci flow blows-up in magnitude at a finite-time singularity is proven. Chapter 6 is on the “Fisher information” functional, leading to the formulation of the Ricci flow as a gradient flow, and on the classical entropy.
Chapter 7 explains the concept of “Cheeger-Gromov” convergence of a sequence of smooth, complete, pointed Riemannian manifolds \((M_i,g_i,p_i)\), where \(p_i\in M_i\), and – more generally – that of convergence of a sequence of smooth families \((M_i,(g_i)_t,p_i)\) of such manifolds, for \(t\) in some open interval, and states a compactness theorem for a sequence of smooth, complete, pointed Riemannian manifolds (all of the same dimension) as well as for a sequence of Ricci flows [see J. Cheeger, M. Gromov, J. Differ. Geom. 23, 309–346 (1986; Zbl 0606.53028); R. S. Hamilton, Am. J. Math. 117, 545–572 (1995; Zbl 0840.53029)]. Rescalings of Ricci flows near their singularities is analyzed.
Chapter 8 is on Perelman’s \(\mathcal{W}\) entropy functional and its applications. Among them is the possibility to prove the lower bound estimate on the injectivity radius stated in the previous chapter. Also, the discussion of blowing up Ricci flows near singularities, that has begun in the previous chapter, is continued.
Chapter 9 develops a general machinery, involving the Uhlenbeck trick, for constraining the evolution of the Einstein tensor field \(E:=- \text{Ric}+{\frac{R}{2}g}\) on a three-dimensional closed manifold \(M\), which obeys a nonlinear heat equation. This is used to prove, under these assumptions on \(M\), that the condition \(\text{Ric}>0\) is preserved under the Ricci flow and that the Ricci flow with \(Ric>0\) looks “round” where \(R\) is large.
Chapter 10 contains a modern proof of Hamilton’s theorem mentioned at the beginning and comments beyond the case \(\text{Ric}>0\), including the enunciation of a curvature pinching result of R. S. Hamilton [Surv. Differ. Geom., Suppl. J. Differ. Geom. 2, 7–136 (1995; Zbl 0867.53030)] and T. Ivey [Differ. Geom. Appl. 3, 301–307 (1993; Zbl 0788.53034)]. The book ends with an appendix on connected sums and with references including 43 items.
The author assembled a collection of very interesting results, providing a polished presentation that includes some proofs of well-known facts for which there is no good reference, and making this fascinating material accessible to the nonexpert reader. His book is good for self-study, and good for use in a graduate level course. In view of the new ideas and great activity in this theory, the author is to be commended for bringing together some of the major threads of the subject in a very readable account.