The Ricci Flow An Introduction

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An Introduction to the Kähler-Ricci Flow

Sebastien Boucksom,Philippe Eyssidieux,Vincent Guedj
An Introduction to the K hler Ricci Flow Book PDF

Author : Sebastien Boucksom,Philippe Eyssidieux,Vincent Guedj
Publisher : Springer
Page : 333 pages
File Size : 53,7 Mb
Release : 2013-10-02
Category : Mathematics
ISBN : 9783319008196

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An Introduction to the Kähler-Ricci Flow by Sebastien Boucksom,Philippe Eyssidieux,Vincent Guedj Pdf

This volume collects lecture notes from courses offered at several conferences and workshops, and provides the first exposition in book form of the basic theory of the Kähler-Ricci flow and its current state-of-the-art. While several excellent books on Kähler-Einstein geometry are available, there have been no such works on the Kähler-Ricci flow. The book will serve as a valuable resource for graduate students and researchers in complex differential geometry, complex algebraic geometry and Riemannian geometry, and will hopefully foster further developments in this fascinating area of research. The Ricci flow was first introduced by R. Hamilton in the early 1980s, and is central in G. Perelman’s celebrated proof of the Poincaré conjecture. When specialized for Kähler manifolds, it becomes the Kähler-Ricci flow, and reduces to a scalar PDE (parabolic complex Monge-Ampère equation). As a spin-off of his breakthrough, G. Perelman proved the convergence of the Kähler-Ricci flow on Kähler-Einstein manifolds of positive scalar curvature (Fano manifolds). Shortly after, G. Tian and J. Song discovered a complex analogue of Perelman’s ideas: the Kähler-Ricci flow is a metric embodiment of the Minimal Model Program of the underlying manifold, and flips and divisorial contractions assume the role of Perelman’s surgeries.

Hamilton’s Ricci Flow

Bennett Chow,Peng Lu,Lei Ni
Hamilton s Ricci Flow Book PDF

Author : Bennett Chow,Peng Lu,Lei Ni
Publisher : American Mathematical Society, Science Press
Page : 648 pages
File Size : 41,5 Mb
Release : 2023-07-13
Category : Mathematics
ISBN : 9781470473693

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Hamilton’s Ricci Flow by Bennett Chow,Peng Lu,Lei Ni Pdf

Ricci flow is a powerful analytic method for studying the geometry and topology of manifolds. This book is an introduction to Ricci flow for graduate students and mathematicians interested in working in the subject. To this end, the first chapter is a review of the relevant basics of Riemannian geometry. For the benefit of the student, the text includes a number of exercises of varying difficulty. The book also provides brief introductions to some general methods of geometric analysis and other geometric flows. Comparisons are made between the Ricci flow and the linear heat equation, mean curvature flow, and other geometric evolution equations whenever possible. Several topics of Hamilton's program are covered, such as short time existence, Harnack inequalities, Ricci solitons, Perelman's no local collapsing theorem, singularity analysis, and ancient solutions. A major direction in Ricci flow, via Hamilton's and Perelman's works, is the use of Ricci flow as an approach to solving the Poincaré conjecture and Thurston's geometrization conjecture.

The Ricci Flow in Riemannian Geometry

Ben Andrews,Christopher Hopper
The Ricci Flow in Riemannian Geometry Book PDF

Author : Ben Andrews,Christopher Hopper
Publisher : Springer Science & Business Media
Page : 306 pages
File Size : 43,8 Mb
Release : 2011
Category : Mathematics
ISBN : 9783642162855

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The Ricci Flow in Riemannian Geometry by Ben Andrews,Christopher Hopper Pdf

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.

The Ricci Flow: Techniques and Applications

Anonim
The Ricci Flow Techniques and Applications Book PDF

Author : Anonim
Publisher : American Mathematical Soc.
Page : 562 pages
File Size : 52,5 Mb
Release : 2007-04-11
Category : Mathematics
ISBN : 9780821839461

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The Ricci Flow: Techniques and Applications by Anonim Pdf

This book gives a presentation of topics in Hamilton's Ricci flow for graduate students and mathematicians interested in working in the subject. The authors have aimed at presenting technical material in a clear and detailed manner. In this volume, geometric aspects of the theory have been emphasized. The book presents the theory of Ricci solitons, Kahler-Ricci flow, compactness theorems, Perelman's entropy monotonicity and no local collapsing, Perelman's reduced distance function and applications to ancient solutions, and a primer of 3-manifold topology. Various technical aspects of Ricci flow have been explained in a clear and detailed manner. The authors have tried to make some advanced material accessible to graduate students and nonexperts. The book gives a rigorous introduction to Perelman's work and explains technical aspects of Ricci flow useful for singularity analysis. Throughout, there are appropriate references so that the reader may further pursue the statements and proofs of the various results.

Ricci Flow and the Poincare Conjecture

John W. Morgan,Gang Tian
Ricci Flow and the Poincare Conjecture Book PDF

Author : John W. Morgan,Gang Tian
Publisher : American Mathematical Soc.
Page : 586 pages
File Size : 48,8 Mb
Release : 2007
Category : Mathematics
ISBN : 0821843281

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Ricci Flow and the Poincare Conjecture by John W. Morgan,Gang Tian Pdf

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).

Generalized Ricci Flow

Mario Garcia-Fernandez,Jeffrey Streets
Generalized Ricci Flow Book PDF

Author : Mario Garcia-Fernandez,Jeffrey Streets
Publisher : American Mathematical Soc.
Page : 248 pages
File Size : 47,9 Mb
Release : 2021-04-06
Category : Education
ISBN : 9781470462581

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Generalized Ricci Flow by Mario Garcia-Fernandez,Jeffrey Streets Pdf

The generalized Ricci flow is a geometric evolution equation which has recently emerged from investigations into mathematical physics, Hitchin's generalized geometry program, and complex geometry. This book gives an introduction to this new area, discusses recent developments, and formulates open questions and conjectures for future study. The text begins with an introduction to fundamental aspects of generalized Riemannian, complex, and Kähler geometry. This leads to an extension of the classical Einstein-Hilbert action, which yields natural extensions of Einstein and Calabi-Yau structures as ‘canonical metrics’ in generalized Riemannian and complex geometry. The book then introduces generalized Ricci flow as a tool for constructing such metrics and proves extensions of the fundamental Hamilton/Perelman regularity theory of Ricci flow. These results are refined in the setting of generalized complex geometry, where the generalized Ricci flow is shown to preserve various integrability conditions, taking the form of pluriclosed flow and generalized Kähler-Ricci flow, leading to global convergence results and applications to complex geometry. Finally, the book gives a purely mathematical introduction to the physical idea of T-duality and discusses its relationship to generalized Ricci flow. The book is suitable for graduate students and researchers with a background in Riemannian and complex geometry who are interested in the theory of geometric evolution equations.

The Ricci Flow: An Introduction

Bennett Chow,Dan Knopf
The Ricci Flow An Introduction Book PDF

Author : Bennett Chow,Dan Knopf
Publisher : American Mathematical Soc.
Page : 342 pages
File Size : 53,5 Mb
Release : 2004
Category : Global differential geometry
ISBN : 9780821835159

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The Ricci Flow: An Introduction by Bennett Chow,Dan Knopf Pdf

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.

Ricci Flow and the Sphere Theorem

Simon Brendle
Ricci Flow and the Sphere Theorem Book PDF

Author : Simon Brendle
Publisher : American Mathematical Soc.
Page : 186 pages
File Size : 51,9 Mb
Release : 2010
Category : Mathematics
ISBN : 9780821849385

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Ricci Flow and the Sphere Theorem by Simon Brendle Pdf

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.

Lectures on the Ricci Flow

Peter Topping
Lectures on the Ricci Flow Book PDF

Author : Peter Topping
Publisher : Cambridge University Press
Page : 124 pages
File Size : 49,5 Mb
Release : 2006-10-12
Category : Mathematics
ISBN : 9780521689472

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Lectures on the Ricci Flow by Peter Topping Pdf

An introduction to Ricci flow suitable for graduate students and research mathematicians.

Hamilton's Ricci Flow

Bennett Chow,Peng Lu,Lei Ni
Hamilton s Ricci Flow Book PDF

Author : Bennett Chow,Peng Lu,Lei Ni
Publisher : American Mathematical Soc.
Page : 656 pages
File Size : 53,8 Mb
Release : 2024-06-23
Category : Mathematics
ISBN : 0821883992

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Hamilton's Ricci Flow by Bennett Chow,Peng Lu,Lei Ni Pdf

Ricci flow is a powerful analytic method for studying the geometry and topology of manifolds. This book is an introduction to Ricci flow for graduate students and mathematicians interested in working in the subject. To this end, the first chapter is a review of the relevant basics of Riemannian geometry. For the benefit of the student, the text includes a number of exercises of varying difficulty. The book also provides brief introductions to some general methods of geometric analysis and other geometric flows. Comparisons are made between the Ricci flow and the linear heat equation, mean curvature flow, and other geometric evolution equations whenever possible. Several topics of Hamilton's program are covered, such as short time existence, Harnack inequalities, Ricci solitons, Perelman's no local collapsing theorem, singularity analysis, and ancient solutions. A major direction in Ricci flow, via Hamilton's and Perelman's works, is the use of Ricci flow as an approach to solving the Poincare conjecture and Thurston's geometrization conjecture.

Collected Papers on Ricci Flow

Huai-Dong Cao
Collected Papers on Ricci Flow Book PDF

Author : Huai-Dong Cao
Publisher : International Pressof Boston Incorporated
Page : 539 pages
File Size : 55,7 Mb
Release : 2003
Category : Mathematics
ISBN : 1571461108

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Collected Papers on Ricci Flow by Huai-Dong Cao Pdf

The Ricci flow is a hot topic at the forefront of mathematics research. This selection of papers on the Riemannian Ricci flow is intended both for the graduate student or researcher unfamiliar with the Ricci flow and for geometers already familiar to the Ricci flow.

Ricci Flow for Shape Analysis and Surface Registration

Wei Zeng,Xianfeng David Gu
Ricci Flow for Shape Analysis and Surface Registration Book PDF

Author : Wei Zeng,Xianfeng David Gu
Publisher : Springer Science & Business Media
Page : 150 pages
File Size : 50,7 Mb
Release : 2013-10-18
Category : Mathematics
ISBN : 9781461487814

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Ricci Flow for Shape Analysis and Surface Registration by Wei Zeng,Xianfeng David Gu Pdf

​Ricci Flow for Shape Analysis and Surface Registration introduces the beautiful and profound Ricci flow theory in a discrete setting. By using basic tools in linear algebra and multivariate calculus, readers can deduce all the major theorems in surface​ Ricci flow by themselves. The authors adapt the Ricci flow theory to practical computational algorithms, apply Ricci flow for shape analysis and surface registration, and demonstrate the power of Ricci flow in many applications in medical imaging, computer graphics, computer vision and wireless sensor network. Due to minimal pre-requisites, this book is accessible to engineers and medical experts, including educators, researchers, students and industry engineers who have an interest in solving real problems related to shape analysis and surface registration.

Extrinsic Geometric Flows

Bennett Chow,Christine Guenther,Mat Langford
Extrinsic Geometric Flows Book PDF

Author : Bennett Chow,Christine Guenther,Mat Langford
Publisher : American Mathematical Soc.
Page : 790 pages
File Size : 44,5 Mb
Release : 2020-05-14
Category : Education
ISBN : 9781470455965

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Extrinsic Geometric Flows by Bennett Chow,Christine Guenther,Mat Langford Pdf

Extrinsic geometric flows are characterized by a submanifold evolving in an ambient space with velocity determined by its extrinsic curvature. The goal of this book is to give an extensive introduction to a few of the most prominent extrinsic flows, namely, the curve shortening flow, the mean curvature flow, the Gauß curvature flow, the inverse-mean curvature flow, and fully nonlinear flows of mean curvature and inverse-mean curvature type. The authors highlight techniques and behaviors that frequently arise in the study of these (and other) flows. To illustrate the broad applicability of the techniques developed, they also consider general classes of fully nonlinear curvature flows. The book is written at the level of a graduate student who has had a basic course in differential geometry and has some familiarity with partial differential equations. It is intended also to be useful as a reference for specialists. In general, the authors provide detailed proofs, although for some more specialized results they may only present the main ideas; in such cases, they provide references for complete proofs. A brief survey of additional topics, with extensive references, can be found in the notes and commentary at the end of each chapter.

The Ricci Flow: Techniques and Applications

Bennett Chow,Sun-Chin Chu,David Glickenstein,Christine Guenther,James Isenberg,Tom Ivey,Dan Knopf,Peng Lu,Feng Luo,Lei Ni
The Ricci Flow Techniques and Applications Book PDF

Author : Bennett Chow,Sun-Chin Chu,David Glickenstein,Christine Guenther,James Isenberg,Tom Ivey,Dan Knopf,Peng Lu,Feng Luo,Lei Ni
Publisher : American Mathematical Soc.
Page : 542 pages
File Size : 48,5 Mb
Release : 2010-04-21
Category : Mathematics
ISBN : 9780821846612

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The Ricci Flow: Techniques and Applications by Bennett Chow,Sun-Chin Chu,David Glickenstein,Christine Guenther,James Isenberg,Tom Ivey,Dan Knopf,Peng Lu,Feng Luo,Lei Ni Pdf

The Ricci flow uses methods from analysis to study the geometry and topology of manifolds. With the third part of their volume on techniques and applications of the theory, the authors give a presentation of Hamilton's Ricci flow for graduate students and mathematicians interested in working in the subject, with an emphasis on the geometric and analytic aspects. The topics include Perelman's entropy functional, point picking methods, aspects of Perelman's theory of $\kappa$-solutions including the $\kappa$-gap theorem, compactness theorem and derivative estimates, Perelman's pseudolocality theorem, and aspects of the heat equation with respect to static and evolving metrics related to Ricci flow. In the appendices, we review metric and Riemannian geometry including the space of points at infinity and Sharafutdinov retraction for complete noncompact manifolds with nonnegative sectional curvature. As in the previous volumes, the authors have endeavored, as much as possible, to make the chapters independent of each other. The book makes advanced material accessible to graduate students and nonexperts. It includes a rigorous introduction to some of Perelman's work and explains some technical aspects of Ricci flow useful for singularity analysis. The authors give the appropriate references so that the reader may further pursue the statements and proofs of the various results.

The Ricci Flow

Bennett Chow
The Ricci Flow Book PDF

Author : Bennett Chow
Publisher : American Mathematical Society(RI)
Page : 562 pages
File Size : 55,7 Mb
Release : 2007
Category : Global differential geometry
ISBN : 1470413620

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The Ricci Flow by Bennett Chow Pdf

Geometric analysis has become one of the most important tools in geometry and topology. In their books on the Ricci flow, the authors reveal the depth and breadth of this flow method for understanding the structure of manifolds. With the present book, the authors focus on the analytic aspects of Ricci flow.