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Ricci Flow And The Sphere Theorem

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Ricci Flow And The Sphere Theorem

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Author : Simon Brendle
language : en
Publisher: American Mathematical Soc.
Release Date : 2010

Ricci Flow And The Sphere Theorem written by Simon Brendle and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010 with Mathematics categories.

Deals with the Ricci flow, and the convergence theory for the Ricci flow. This title focuses on preserved curvature conditions, such as positive isotropic curvature. It is suitable for graduate students and researchers.

The Ricci Flow In Riemannian Geometry

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Author : Ben Andrews
language : en
Publisher: Springer Science & Business Media
Release Date : 2011

The Ricci Flow In Riemannian Geometry written by Ben Andrews and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011 with Mathematics categories.

This book focuses on Hamilton's Ricci flow, beginning with a detailed discussion of the required aspects of differential geometry, progressing through existence and regularity theory, compactness theorems for Riemannian manifolds, and Perelman's noncollapsing results, and culminating in a detailed analysis of the evolution of curvature, where recent breakthroughs of Böhm and Wilking and Brendle and Schoen have led to a proof of the differentiable 1/4-pinching sphere theorem.

Ricci Flow And The Poincare Conjecture

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Author : John W. Morgan
language : en
Publisher: American Mathematical Soc.
Release Date : 2007

Ricci Flow And The Poincare Conjecture written by John W. Morgan and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007 with Mathematics categories.

For over 100 years the Poincare Conjecture, which proposes a topological characterization of the 3-sphere, has been the central question in topology. Since its formulation, it has been repeatedly attacked, without success, using various topological methods. Its importance and difficulty were highlighted when it was chosen as one of the Clay Mathematics Institute's seven Millennium Prize Problems. in 2002 and 2003 Grigory Perelman posted three preprints showing how to use geometric arguments, in particular the Ricci flow as introduced and studied by Hamilton, to establish the Poincare Conjecture in the affirmative. This book provides full details of a complete proof of the Poincare Conjecture following Perelman's three preprints. After a lengthy introduction that outlines the entire argument, the book is divided into four parts. The first part reviews necessary results from Riemannian geometry and Ricci flow, including much of Hamilton's work. The second part starts with Perelman's length function, which is used to establish crucial non-collapsing theorems. Then it discusses the classification of non-collapsed, ancient solutions to the Ricci flow equation. The third part concerns the existence of Ricci flow with surgery for all positive time and an analysis of the topological and geometric changes introduced by surgery. The last part follows Perelman's third preprint to prove that when the initial Riemannian 3-manifold has finite fundamental group, Ricci flow with surgery becomes extinct after finite time. The proofs of the Poincare Conjecture and the closely related 3-dimensional spherical space-form conjectu The existence of Ricci flow with surgery has application to 3-manifolds far beyond the Poincare Conjecture. It forms the heart of the proof via Ricci flow of Thurston's Geometrization Conjecture. Thurston's Geometrization Conjecture, which classifies all compact 3-manifolds, will be the subject of a follow-up article. The organization of the material in this book differs from that given by Perelman. From the beginning the authors present all analytic and geometric arguments in the context of Ricci flow with surgery. in addition, the fourth part is a much-expanded version of Perelman's third preprint; it gives the first complete and detailed proof of the finite-time extinction theorem. With the large amount of background material that is presented and the detailed versions of the central arguments, this book is suitable for all mathematicians from advanced graduate students to specialists in geometry and topology. Clay Mathematics Institute Monograph Series The Clay Mathematics Institute Monograph Series publishes selected expositions of recent developments, both in emerging areas and in older subjects transformed by new insights or unifying ideas. Information for our distributors: Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA).

Ricci Flow And The Sphere Theorem

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Author : Simon Brendle
language : en
Publisher: American Mathematical Society
Release Date : 2024-11-06

Ricci Flow And The Sphere Theorem written by Simon Brendle and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-11-06 with Mathematics categories.

In 1982, R. Hamilton introduced a nonlinear evolution equation for Riemannian metrics with the aim of finding canonical metrics on manifolds. This evolution equation is known as the Ricci flow, and it has since been used widely and with great success, most notably in Perelman's solution of the Poincar‚ conjecture. Furthermore, various convergence theorems have been established. This book provides a concise introduction to the subject as well as a comprehensive account of the convergence theory for the Ricci flow. The proofs rely mostly on maximum principle arguments. Special emphasis is placed on preserved curvature conditions, such as positive isotropic curvature. One of the major consequences of this theory is the Differentiable Sphere Theorem: a compact Riemannian manifold, whose sectional curvatures all lie in the interval (1,4], is diffeomorphic to a spherical space form. This question has a long history, dating back to a seminal paper by H. E. Rauch in 1951, and it was resolved in 2007 by the author and Richard Schoen. This text originated from graduate courses given at ETH Z�rich and Stanford University, and it is directed at graduate students and researchers. The reader is assumed to be familiar with basic Riemannian geometry, but no previous knowledge of Ricci flow is required.

The Ricci Flow In Riemannian Geometry

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Author : Ben Andrews
language : en
Publisher: Springer
Release Date : 2010-11-09

The Ricci Flow In Riemannian Geometry written by Ben Andrews and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-11-09 with Mathematics categories.

The Ricci Flow An Introduction

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Author : Bennett Chow
language : en
Publisher: American Mathematical Soc.
Release Date : 2004

The Ricci Flow An Introduction written by Bennett Chow and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004 with Mathematics categories.

The Ricci flow is a powerful technique that integrates geometry, topology, and analysis. Intuitively, the idea is to set up a PDE that evolves a metric according to its Ricci curvature. The resulting equation has much in common with the heat equation, which tends to 'flow' a given function to ever nicer functions. By analogy, the Ricci flow evolves an initial metric into improved metrics. Richard Hamilton began the systematic use of the Ricci flow in the early 1980s and applied it in particular to study 3-manifolds. Grisha Perelman has made recent breakthroughs aimed at completing Hamilton's program. The Ricci flow method is now central to our understanding of the geometry and topology of manifolds.This book is an introduction to that program and to its connection to Thurston's geometrization conjecture. The authors also provide a 'Guide for the hurried reader', to help readers wishing to develop, as efficiently as possible, a nontechnical appreciation of the Ricci flow program for 3-manifolds, i.e., the so-called 'fast track'. The book is suitable for geometers and others who are interested in the use of geometric analysis to study the structure of manifolds. "The Ricci Flow" was nominated for the 2005 Robert W. Hamilton Book Award, which is the highest honor of literary achievement given to published authors at the University of Texas at Austin.

Comparison Geometry

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Author : Karsten Grove
language : en
Publisher: Cambridge University Press
Release Date : 1997-05-13

Comparison Geometry written by Karsten Grove and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997-05-13 with Mathematics categories.

This is an up to date work on a branch of Riemannian geometry called Comparison Geometry.

The Poincar Conjecture

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Author : Donal O'Shea
language : en
Publisher: Penguin UK
Release Date : 2008-10-30

The Poincar Conjecture written by Donal O'Shea and has been published by Penguin UK this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-10-30 with Science categories.

The Poincaré Conjecture tells the story behind one of the world’s most confounding mathematical theories. Formulated in 1904 by Henri Poincaré, his Conjecture promised to describe the very shape of the universe, but remained unproved until a huge prize was offered for its solution in 2000. Six years later, an eccentric Russian mathematician had the answer. Here, Donal O’Shea explains the maths behind the Conjecture and its proof, and illuminates the curious personalities surrounding this perplexing conundrum, along the way taking in a grand sweep of scientific history from the ancient Greeks to Christopher Columbus. This is an enthralling tale of human endeavour, intellectual brilliance and the thrill of discovery.

Ricci Flow And A Sphere Theorem

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Author : Shaochuang Huang
language : en
Publisher:
Release Date : 2013

Ricci Flow And A Sphere Theorem written by Shaochuang Huang and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013 with Ricci flow categories.

In this text, we present some background materials for the Ricci flow, including curvature evolution under the Ricci flow, short-time existence, uniqueness and higher derivatives estimate for curvature and tensor. We also focus on the maximum principle and convergence criterion for the Ricci flow. The fact thatnonnegative isotropic curvature is preserved under the Ricci flow will be showed. Finally, we complete the proof of the differentiable sphere theorem using a family of invariant cones which was constructed by Böhm and Wilking.

Ricci Flow And The Sphere Theorem

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