The Ricci Flow: Techniques and Applications
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The Ricci Flow: Techniques and Applications: Part IV: Long-Time Solutions and Related Topics

The Ricci Flow: Techniques and Applications: Part IV: Long-Time Solutions and Related Topics

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About the Book

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.

Table of Contents:
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


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Product Details
  • ISBN-13: 9780821849910
  • Publisher: American Mathematical Society
  • Publisher Imprint: American Mathematical Society
  • Height: 254 mm
  • No of Pages: 374
  • Weight: 855 gr
  • ISBN-10: 0821849913
  • Publisher Date: 30 Dec 2015
  • Binding: Hardback
  • Language: English
  • Sub Title: Part IV: Long-Time Solutions and Related Topics
  • Width: 178 mm


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