This is the first volume of a two-part sequel of The Ricci Flow: An Introduction by the same authors, which laid out the foundations for the study of Richard Hamilton's Ricci flow, an evolution equation which deforms Riemannian metrics by evolving them in the direction of minus the Ricci tensor. In this volume, they continue to explore the fundamental properties of the Ricci flow, with particular emphasis on their application to the study of singularities. The volume's eight chapters discuss Ricci solitons; K�hler-Ricci flow and K�hler- Ricci solitons; the compactness theorem for Ricci flow and its proof; energy, monotonicity, and breathers; entropy and non local collapsing; the reduced distance and its applications; and the basic topology of 3-manifolds. Annotation �2007 Book News, Inc., Portland, OR (booknews.com)

Review

"Just like in the first part this book is meant to be a text book as well as a reference book and it includes exercises as well as the solutions for some of them. The authors took a great care in making this a self contained book which compiles a great amount of data related to Ricci flow, organizing the information in a very clear way giving very often information on the content of each chapter and relating it to other parts of the book. An example is the sumary given in the preface, another more detailed sumary given in a special kind of chapter zero, also in the beginning of each chapter and sometimes introducing each section. Besides all this, each chapter includes a special section in the end with notes and commentaries. This approach gives the reader permanently a global view on the subject while the details are being explained. The book also includes a large number of exercises very often with suggestions for its resolutions." ---- Zentralblatt MATH

"This book and this series is necessarily quite technical in parts, but it does reveal the full beauty and technical details of the Ricci flow. The authors should be congratulated on presenting such material in a very readable style, creating an excellent resource with thorough discussion of many aspects of Ricci flow for the benefit of mathematical researchers today and in the future." ---- Mathematical Reviews

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