zbMATH — the first resource for mathematics

Hamilton’s Ricci flow. (English) Zbl 1118.53001
Graduate Studies in Mathematics 77. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-4231-5/hbk). xxxvi, 608 p. (2006).
Ricci flow, exposed in [B. Chow, D. Knopf, The Ricci flow: an introduction. Mathematical Surveys and Monographs 110. Providence, RI: American Mathematical Society (AMS) (2004; Zbl 1086.53085)], is a geometric and analytic evolution equation, related to physical reality. The present book begins with an introduction to Riemannian geometry, followed by some fundamentals of the Ricci flow equation. A proof of Hamilton’s classification of closed 3-manifolds with positive Ricci curvature using Ricci flow is given. Ricci solitons, homogeneous solutions and other special solutions are studied. Monotonicity formulas, which yield isoperimetric and volume ratio estimate are also discussed. The analytic results and techniques are collected since they are useful for singularity analysis. The chapter “High-dimensional and noncompact Ricci flow”, the spherical space form theorem of Huisken, Margerin and Nishikawa are exposed.
To study singularities one takes dilations about sequences of points and times where the time tends to the singularity time. The limit solutions of such sequences, if they exist, are ancient solutions. It is useful to distinguish singular and ancient solutions according to the rate of blow up of the curvature. Various differential Harnack estimates are discussed. At the end, there are various space-time geometries which are rather similar, culminating in Perelman’s metric on the product of space-time with large-dimensional and large radius sphere. The book contains a number of exercises, two appendices and a bibliography including a number of evolution equations, geometric analysis and related areas.

53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
58J35 Heat and other parabolic equation methods for PDEs on manifolds
35K55 Nonlinear parabolic equations
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions