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The Ricci flow: an introduction. (English) Zbl 1086.53085
Mathematical Surveys and Monographs 110. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3515-7/hbk). xii, 325 p. (2004).
The book under review is the first volume in a series comprising an introduction to the Ricci flow, and to the program posed by Richard Hamilton for using the Ricci flow to resolve Thurston’s geometrization conjecture for closed three-manifolds. The Ricci flow is a geometric evolution equation in which the metric of a smooth Riemannian manifold \((M^{n},g_(0))\) evolves by its Ricci tensor: \[ {\partial \over \partial t}g = -2Rc. \] The field is of great currency, in part because of Grisha Perelman’s recent work on completing the proof of the conjecture [“The entropy formula for the Ricci flow and its geometric applications”, arXiv:math.DG/0211159, “Ricci flow with surgery on three-manifolds”, arXiv: math.DG/0303190, “Finite extinction time for the solutions to the Ricci flow on certain three-manifolds”, arXiv: math.DG/0307245].
R. S. Hamilton first studied the Ricci flow in his seminal paper [J. Differ. Geom. 17, No. 2, 255–306 (1982; Zbl 0504.53034)], in which he proved that any closed three-dimensional manifold of positive Ricci curvature is diffeomorphic to a spherical space form. Since then further essential work in the field (e.g. on maximum principles and differential Harnack inequalities) has had an enormous impact on geometric analysis in general, and in particular on geometric evolution equations.
The background required for the book is a basic knowledge of Riemannian geometry. An understanding of algebraic topology and partial differential equations is useful, although the authors precisely define the terms and develop the material used from these fields. This volume is well-written, and the topics are well-motivated and presented with clarity and insight. The proofs are clear yet detailed so that the material is accessible to a wide audience as well as specialists. Gaps in arguments in the literature are filled in. The topics in this volume are:
Chapter 1 (The Ricci flow of special geometries): In the first two chapters the authors motivate the study of the Ricci flow, and develop intuition for it by considering the behavior of special solutions. The first chapter includes a discussion of Thurston’s geometrization conjecture, and the evolution of homogeneous metrics [J. Isenberg and M. Jackson, J. Differ. Geom. 35, No. 3, 723–741 (1992; Zbl 0808.53044)]. The geometrization conjecture roughly states that every closed three-manifold can be canonically decomposed into pieces, each of which admits a unique homogeneous geometry.
Chapter 2 (Special and limit solutions): The authors introduce self-similar solutions, or Ricci solitons. These are solutions that change only by diffeomorphism and re-scaling, and so are fixed points of the flow when it is considered as a dynamical system on the space of smooth Riemannian metrics modulo diffeomorphisms. Examples of solutions that exist for infinite time intervals are then presented. These are important to understanding singularity formation, since they can arise as limits of dilations about singularities, or “singularity models” (Chapters 8 and 9). “Degenerate neck-pinch” singularities are discussed, and “neck-pinch” solutions are constructed rigorously.
Chapter 3 (Short time existence): The Ricci flow is a weakly parabolic system and so short time existence, originally proved by Hamilton using the Nash-Moser implicit function theorem, is not provided by standard parabolic theory. In this chapter the authors follow DeTurck’s elegant method of proof, in which he shows that the Ricci flow is equivalent to a strictly parabolic system. A section relating this to the harmonic map heat flow is included, as is a section explaining how the Ricci flow may be regarded as a heat equation for the metric.
Chapter 4 (Maximum principles): Maximum principles for functions and tensors are among the most important tools in geometric evolution equations. This chapter begins with simple results to elucidate the arguments used in the derivation of maximum principles, and then progresses to more advanced theorems, including Hamilton’s crucial maximum principles for tensors.
Chapter 5 (The Ricci flow on surfaces): In this chapter the authors present the Ricci flow on closed surfaces. The main result is that a solution of the normalized Ricci flow on closed two-dimensional manifolds exists for all time, and converges uniformly in any \(C^{k}\) norm to a smooth metric of constant curvature that is conformal to the original metric. Solitons play a key role in the proof, as do entropy and the differential Harnack inequality.
Chapter 6: Three-manifolds of positive Ricci curvature Hamilton’s original result for closed three-manifolds of positive Ricci curvature is presented. The evolution equations for the connection and curvatures are computed, and coordinates are constructed via a moving orthonormal frame field. Ideas essential to the rest of the study of Ricci flow appear, including pinching estimates, the maximum principe for tensors, and derivative estimates. The maximum principle for tensors is used to prove that the eigenvalues of the Ricci tensor approach each other at points of large scalar curvature. Estimates of the gradient of the scalar curvature are then used to compare curvatures at different points. One first works with the unnormalized flow, and then argues that a priori estimstes can be obtained to achieve results for the normalized flow.
Chapter 7 (Derivative estimates): By the smoothing properties of parabolic equations, one expects that bounds on the geometry of \(g_{0}\) should induce bounds on the geometry of the solutions \(g(t)\) to the Ricci flow. In this chapter, the authors present Bando’s and (independently) Shi’s work [S. Bando, Math. Z. 195, No. 1, 93–97 (1987; Zbl 0606.58051); W.-X. Shi, J. Differ. Geom. 30, No. 1, 223–301 (1989; Zbl 0676.53044), J. Differ. Geom. 30, No. 2, 303–394 (1989; Zbl 0686.53037)], in which global short time estimates on all derviatives of the curvature are established. This allows one to prove that a solution to the flow exists as long as the curvatures remain bounded. The chapter concludes with a statement of the Compactness Theorem (to be proved in a successor volune), which is used to show the existence of “singularity models”.
Chapter 8 and 9 (Singularities and the limits of their dilations; Type I singularities): Here the authors classify maximal time solutions to the Ricci flow and introduce parabolic dilation about singularities to derive “singularity models”. Consider a solution that exists up to a maximal time \(T \in (0,\infty]\). To implement Hamilton’s program for resolving Thurston’s geometrization conjecture, one needs to understand the geometry of the solution shortly before the singularity occurs in order to perform geometric-topological surgeries to move beyond the singularity. The method of parabolic dilation about a singularity iinvolves choosing a suitable sequence of points and times that approach the singularity, then dilating in space and time and translating in time to find a sequence of solutions to the flow. Their limit can yield information about the original solution prior to the formation of the singularity. In Chapter 9, the authors explore why singularities in dimension three are tractable, and why one expects to see a “neck” – a piece of the manifold close to a quotient of the shrinking round cylinder \((R \times S^{2}, g(t))\) – devleop near a typical singularity.
Appendices: In Ricci flow calculations in local coordinates are used extensively. In the first appendix, the authors review differential operators on tensors and forms, the Laplacian (rough, Hodge-de Rham, Lichnerowicz), notation for higher derivatives, and formulas for commuting derivatives. Comparison principles are also used throughout the subject, and the authors collect some standard ideas in local geometry (conjugate points, Jacobi fields, Rauch comparison theorem). They further discuss the injectivity radius and lifts of the metric by the exponential map, the Laplacian and Toponogov comparison theorems, and Busemann functions.

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
35K55 Nonlinear parabolic equations
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
58J35 Heat and other parabolic equation methods for PDEs on manifolds
57M50 General geometric structures on low-dimensional manifolds