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)

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