Ricci 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.
Rezensionen / Stimmen
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
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Höhe: 254 mm
Breite: 178 mm
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ISBN-13
978-0-8218-4991-0 (9780821849910)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Klassifikation
Bennett 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.
Noncompact 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
Index
