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OverviewRicci flow is a powerful technique using a heat-type equation to deform Riemannian metrics on manifolds to better metrics in the search for geometric decompositions. With the fourth part of their volume on techniques and applications of the theory, the authors discuss long-time solutions of the Ricci flow and related topics. In dimension 3, Perelman completed Hamilton's program to prove Thurston's geometrization conjecture. In higher dimensions the Ricci flow has remarkable properties, which indicates its usefulness to understand relations between the geometry and topology of manifolds. This book discusses recent developments on gradient Ricci solitons, which model the singularities developing under the Ricci flow. In the shrinking case there is a surprising rigidity which suggests the likelihood of a well-developed structure theory. A broader class of solutions is ancient solutions; the authors discuss the beautiful classification in dimension 2. In higher dimensions they consider both ancient and singular Type I solutions, which must have shrinking gradient Ricci soliton models. Next, Hamilton's theory of 3-dimensional nonsingular solutions is presented, following his original work. Historically, this theory initially connected the Ricci flow to the geometrization conjecture. From a dynamical point of view, one is interested in the stability of the Ricci flow. The authors discuss what is known about this basic problem. Finally, they consider the degenerate neckpinch singularity from both the numerical and theoretical perspectives. This book makes advanced material accessible to researchers and graduate students who are interested in the Ricci flow and geometric evolution equations and who have a knowledge of the fundamentals of the Ricci flow. Full Product DetailsAuthor: Bennett Chow , Sun-Chin Chu , David Glickenstein , Christine GuentherPublisher: American Mathematical Society Imprint: American Mathematical Society Weight: 0.855kg ISBN: 9780821849910ISBN 10: 0821849913 Pages: 374 Publication Date: 30 December 2015 Audience: Professional and scholarly , Professional & Vocational Format: Hardback Publisher's Status: Active Availability: In Print This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us. Table of ContentsNoncompact gradient Ricci solitons Special ancient solutions Compact 2-dimensional ancient solutions Type I singularities and ancient solutions Hyperbolic geometry and 3-manifolds Nonsingular solutions on closed 3-manifolds Noncompact hyperbolic limits Constant mean curvature surfaces and harmonic maps by IFT Stability of Ricci flow Type II singularities and degenerate neckpinches Implicit function theorem Bibliography IndexReviewsThis book concludes a long series of carefully written and extremely detailed textbooks on the Ricci flow, which have instantly become mandatory reading for any graduate student who is interested in doing research in this field. They are also an excellent resource for established researchers in this and neighboring fields. -Valentino Tosatti, Zentralblatt MATH This book concludes a long series of carefully written and extremely detailed textbooks on the Ricci flow, which have instantly become mandatory reading for any graduate student who is interested in doing research in this field. They are also an excellent resource for established researchers in this and neighboring fields. - Valentino Tosatti, Zentralblatt MATH "This book concludes a long series of carefully written and extremely detailed textbooks on the Ricci flow, which have instantly become mandatory reading for any graduate student who is interested in doing research in this field. They are also an excellent resource for established researchers in this and neighboring fields."" - Valentino Tosatti, Zentralblatt MATH" Author InformationBennett Chow, University of California, San Diego, La Jolla, CA, USA. Sun-Chin Chu, National Chung Cheng University, Chia-Yi, Taiwan. David Glickenstein, University of Arizona, Tucson, AZ, USA. Christine Guenther, Pacific University, Forest Grove, OR, USA. James Isenberg, University of Oregon, Eugene, OR, USA. Tom Ivey, The College of Charleston, SC, USA. Dan Knopf, University of Texas at Austin, TX, USA. Peng Lu, University of Oregon, Eugene, OR, USA. Feng Luo, Rutgers University, Piscataway, NJ, USA. Lei Ni, University of California, San Diego, La Jolla, CA, USA. Tab Content 6Author Website:Countries AvailableAll regions |