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Author : Jean Carlos Cortissoz
Publisher :
ISBN 13 :
Total Pages : 202 pages
Book Rating : 4.E/5 ( download)
Download or read book On the Ricci Flow in Rotationally Symmetric Manifolds with Boundary written by Jean Carlos Cortissoz and published by . This book was released on 2004 with total page 202 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Author : Bennett Chow
Publisher : American Mathematical Society, Science Press
ISBN 13 : 1470473690
Total Pages : 648 pages
Book Rating : 4.4/5 (74 download)
Download or read book Hamilton’s Ricci Flow written by Bennett Chow and published by American Mathematical Society, Science Press. This book was released on 2023-07-13 with total page 648 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Author : César Augusto Reyes Castellanos
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (115 download)
Download or read book Ricci Flow on the Cylinder and Stability of Geometric Flows on the Circle written by César Augusto Reyes Castellanos and published by . This book was released on 2019 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: "This thesis has two parts. In the first part we study the Ricci flow on a cylinder with boundary, endowed with an initial metric such that the scalar curvature is negative and the geodesic curvature of the boundary is positive. We proved that under the unnormalized Ricci flow the area remains finite in any finite interval of time, the normalized flow exists for all time if and only if the unnormalized flow so does, and if the initial metric is rotationally symmetric then the scalar curvature remains bounded on bounded intervals of time. In the second part, we study the stability of the circle in the case of some geometric flows and we give some estimates on the convergence toward the unit circle for solutions to these flows, assuming that the initial condition is near a circle. Finally, we give some examples where our main theorem can be applied."-- Tomado del Formato de Documento de Grado.
Author : Bennett Chow
Publisher : American Mathematical Society(RI)
ISBN 13 : 9781470413620
Total Pages : 562 pages
Book Rating : 4.4/5 (136 download)
Download or read book The Ricci Flow written by Bennett Chow and published by American Mathematical Society(RI). This book was released on 2007 with total page 562 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Author : John W. Morgan
Publisher : American Mathematical Soc.
ISBN 13 : 9780821843284
Total Pages : 586 pages
Book Rating : 4.8/5 (432 download)
Download or read book Ricci Flow and the Poincare Conjecture written by John W. Morgan and published by American Mathematical Soc.. This book was released on 2007 with total page 586 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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).
Author : Bennett Chow
Publisher : American Mathematical Soc.
ISBN 13 : 0821835157
Total Pages : 342 pages
Book Rating : 4.8/5 (218 download)
Download or read book The Ricci Flow: An Introduction written by Bennett Chow and published by American Mathematical Soc.. This book was released on 2004 with total page 342 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Author : Bennett Chow
Publisher : American Mathematical Soc.
ISBN 13 : 0821849913
Total Pages : 397 pages
Book Rating : 4.8/5 (218 download)
Download or read book The Ricci Flow: Techniques and Applications written by Bennett Chow and published by American Mathematical Soc.. This book was released on 2015-10-19 with total page 397 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Author : Sebastien Boucksom
Publisher : Springer
ISBN 13 : 3319008196
Total Pages : 342 pages
Book Rating : 4.3/5 (19 download)
Download or read book An Introduction to the Kähler-Ricci Flow written by Sebastien Boucksom and published by Springer. This book was released on 2013-10-02 with total page 342 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Author : Bennett Chow
Publisher : American Mathematical Soc.
ISBN 13 : 0821846612
Total Pages : 542 pages
Book Rating : 4.8/5 (218 download)
Download or read book The Ricci Flow: Techniques and Applications written by Bennett Chow and published by American Mathematical Soc.. This book was released on 2010-04-21 with total page 542 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Author : Bin Guo
Publisher :
ISBN 13 :
Total Pages : 129 pages
Book Rating : 4.:/5 (922 download)
Download or read book Some Parabolic and Elliptic Problems in Complex Riemannian Geometry written by Bin Guo and published by . This book was released on 2015 with total page 129 pages. Available in PDF, EPUB and Kindle. Book excerpt: This dissertation consists of three parts, the first one is on the blow-up behavior of K"ahler Ricci flow on $cp^n$ blown-up at one point, and the second one on the convergence of K"ahler Ricci flow on minimal projective manifolds of general type, and the last one is on the existence of canonical conical K"ahler metrics on toric manifolds. In the first part, we consider the Ricci flow on $cp^n$ blown-up at one point starting with any rotationally symmetric K"ahler metric. We show that if the total volume does not go to zero at the singular time, then any parabolic blow-up limit of the Ricci flow along the exceptional divisor is a non-compact complete shrinking K"ahler Ricci soliton with rotational symmetry on $mathbb C^n$ blown-up at one point, hence the FIK soliton constructed in cite{FIK}. In the second part, we consider the K"ahler Ricci flow on a smooth minimal model of general type, following the ideas of Song (cite{S1, S2}), we show that if the Ricci curvature is uniformly bounded below along the K"ahler-Ricci flow, then the diameter is uniformly bounded. As a corollary we show that under the Ricci curvature lower bound assumption, the Gromov-Hausdorff limit of the flow is homeomorphic to the canonical model of the manifold. Moreover, we will give a purely analytic proof of a recent result of Tosatti-Zhang (cite{TZ}) that if the canonical line bundle $K_X$ is big and nef, but not ample, then the Ricci flow is of Type IIb. In the last part, we give criterion for the existence of toric conical K"ahler-Einstein and K"ahler-Ricci soliton metrics on any toric manifold in relation to the greatest Ricci lower bound and Bakry-Emery-Ricci lower bound. It is shown that any two toric manifolds with the same dimension can be joined by a continuous path of toric manifolds with conical K"ahler-Einstein metrics in the Gromov-Hausdorff topology.
Author : Peter Topping
Publisher : Cambridge University Press
ISBN 13 : 0521689473
Total Pages : 124 pages
Book Rating : 4.5/5 (216 download)
Download or read book Lectures on the Ricci Flow written by Peter Topping and published by Cambridge University Press. This book was released on 2006-10-12 with total page 124 pages. Available in PDF, EPUB and Kindle. Book excerpt: An introduction to Ricci flow suitable for graduate students and research mathematicians.
Author : Tsz Kiu Aaron Chow
Publisher :
ISBN 13 :
Total Pages : 0 pages
Book Rating : 4.:/5 (139 download)
Download or read book Ricci Flow and Positivity of Curvature on Manifolds with Boundary written by Tsz Kiu Aaron Chow and published by . This book was released on 2023 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The results from chapters 2 through 4 will be utilized in proving the Main Theorems in chapter 5. In particular, we construct canonical solutions to the Ricci flow on manifolds with boundary from canonical solutions to the Ricci flow on closed manifolds with Hölder continuous initial data via doubling.
Author : Bennett Chow
Publisher : American Mathematical Soc.
ISBN 13 : 0821844296
Total Pages : 489 pages
Book Rating : 4.8/5 (218 download)
Download or read book The Ricci Flow: Techniques and Applications written by Bennett Chow and published by American Mathematical Soc.. This book was released on 2007 with total page 489 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Author :
Publisher :
ISBN 13 :
Total Pages : 794 pages
Book Rating : 4.F/5 ( download)
Download or read book Dissertation Abstracts International written by and published by . This book was released on 2005 with total page 794 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Author : Manuel Ritoré
Publisher : Springer Science & Business Media
ISBN 13 : 3034602138
Total Pages : 113 pages
Book Rating : 4.0/5 (346 download)
Download or read book Mean Curvature Flow and Isoperimetric Inequalities written by Manuel Ritoré and published by Springer Science & Business Media. This book was released on 2010-01-01 with total page 113 pages. Available in PDF, EPUB and Kindle. Book excerpt: Geometric flows have many applications in physics and geometry. The mean curvature flow occurs in the description of the interface evolution in certain physical models. This is related to the property that such a flow is the gradient flow of the area functional and therefore appears naturally in problems where a surface energy is minimized. The mean curvature flow also has many geometric applications, in analogy with the Ricci flow of metrics on abstract riemannian manifolds. One can use this flow as a tool to obtain classification results for surfaces satisfying certain curvature conditions, as well as to construct minimal surfaces. Geometric flows, obtained from solutions of geometric parabolic equations, can be considered as an alternative tool to prove isoperimetric inequalities. On the other hand, isoperimetric inequalities can help in treating several aspects of convergence of these flows. Isoperimetric inequalities have many applications in other fields of geometry, like hyperbolic manifolds.
Author : Ramesh Sharma
Publisher : Springer Nature
ISBN 13 : 9819992583
Total Pages : 165 pages
Book Rating : 4.8/5 (199 download)
Download or read book Conformal Vector Fields, Ricci Solitons and Related Topics written by Ramesh Sharma and published by Springer Nature. This book was released on 2024-01-19 with total page 165 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an up-to-date introduction to the theory of manifolds, submanifolds, semi-Riemannian geometry and warped product geometry, and their applications in geometry and physics. It then explores the properties of conformal vector fields and conformal transformations, including their fixed points, essentiality and the Lichnerowicz conjecture. Later chapters focus on the study of conformal vector fields on special Riemannian and Lorentzian manifolds, with a special emphasis on general relativistic spacetimes and the evolution of conformal vector fields in terms of initial data. The book also delves into the realm of Ricci flow and Ricci solitons, starting with motivations and basic results and moving on to more advanced topics within the framework of Riemannian geometry. The main emphasis of the book is on the interplay between conformal vector fields and Ricci solitons, and their applications in contact geometry. The book highlights the fact that Nil-solitons and Sol-solitons naturally arise in the study of Ricci solitons in contact geometry. Finally, the book gives a comprehensive overview of generalized quasi-Einstein structures and Yamabe solitons and their roles in contact geometry. It would serve as a valuable resource for graduate students and researchers in mathematics and physics as well as those interested in the intersection of geometry and physics.
Author : Huai-Dong Cao
Publisher :
ISBN 13 :
Total Pages : 366 pages
Book Rating : 4.3/5 (91 download)
Download or read book Geometric Flows written by Huai-Dong Cao and published by . This book was released on 2008 with total page 366 pages. Available in PDF, EPUB and Kindle. Book excerpt:
