The Ricci flow is a powerful technique that integrates geometry, topology, and analysis. Intuitively, the idea is to set up a PDE that evolves a metric according to its Ricci curvature. The resulting equation has much in common with the heat equation, which tends to "flow" a given function to ever nicer functions. By analogy, the Ricci flow evolves an initial metric into improved metrics.

Richard Hamilton began the systematic use of the Ricci flow in the early 1980s and applied it in particular to study 3-manifolds. Grisha Perelman has made recent breakthroughs aimed at completing Hamilton's program.

The Ricci flow method is now central to our understanding of the geometry and topology of manifolds. This book is an introduction to that program and to its connection to Thurston's geometrization conjecture.

The authors also provide a "Guide for the hurried reader", to help readers wishing to develop, as efficiently as possible, a nontechnical appreciation of the Ricci flow program for 3-manifolds, i.e., the so-called "fast track".

The book is suitable for geometers and others who are interested in the use of geometric analysis to study the structure of manifolds.

The Ricci Flow was nominated for the 2005 Robert W. Hamilton Book Award, which is the highest honor of literary achievement given to published authors at the University of Texas at Austin.

Readership

Graduate students and research mathematicians interested in geometric analysis.

Reviews

"Well written ... topics are well-motivated and presented with clarity and insight ... 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."

-- Zentralblatt MATH

Table of Contents

• The Ricci flow of special geometries
• Special and limit solutions
• Short time existence
• Maximum principles
• The Ricci flow on surfaces
• Three-manifolds of positive Ricci curvature
• Derivative estimates
• Singularities and the limits of their dilations
• Type I singularities
• The Ricci calculus
• Some results in comparison geometry
• Bibliography
• Index