The aim of this book is to give an introduction to the subject of Ricci flow on Riemannian manifolds and to the geometrization conjecture, a topic initiated by Hamilton's paper on 3-manifolds with positive Ricci curvature and completed in the recent spectacular developments by G. Perelman. In the introductory chapter, the book reviews from scratch some basic results and facts from Riemannian geometry. The next chapters introduce Ricci flow and discuss the proof of Hamilton's classification of 3-manifolds with positive Ricci curvature together with a presentation of special (e.g. homogeneous, solitonic) solutions. Chapters 5 and 6 start the discussion of various partial differential equations, aspects of the Ricci flow like monotonicity formulas and techniques useful in the analysis of singularities. In chapter 10, the reader can find various differential Harnack estimates allowing the control and comparison of solutions of Ricci flow at different points of the space-time. The last chapter then deals with various implications of Ricci flow towards the existence of special (degenerate, Einstein) metrics. The book is in fact an introduction to the topic of geometry and topology of 3-manifolds via the Ricci flow technique for graduate students and mathematicians interested in the field. It contains many exercises and open problems throughout the book.

Reviewer:

pso