Rauch in , and it was resolved in by the author and Richard Schoen. This text originated from graduate courses given at ETH Zurich 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. Skip to content Close Menu Contact. Release : Category: Mathematics Total Pages : ISBN : Ricci flow is a powerful analytic method for studying the geometry and topology of manifolds.
Release : Category: N. Willenbring Publisher: Springer Reads. Author : N. Geometric analysis. Book Summary: In , 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 Poincare 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. Rauch in , and it was resolved in by the author and Richard Schoen.
This text originated from graduate courses given at ETH Zurich 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. Book Summary: Presenting some impressive recent achievements in differential geometry and topology, this volume focuses on results obtained using techniques based on Ricci flow. These ideas are at the core of the study of differentiable manifolds.
Several very important open problems and conjectures come from this area and the techniques described herein are used to face and solve some of them. Besson , the geometrization of 3-manifolds M. Boileau , the singularities of 3-dimensional Ricci flows C. The lectures will be particularly valuable to young researchers interested in differential manifolds.
Book Summary: 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. Book Summary: 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 s 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.
The book is suitable for geometers and others who are interested in the use of geometric analysis to study the structure of manifolds. Hamilton Book Award, which is the highest honor of literary achievement given to published authors at the University of Texas at Austin. Book Summary: 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. Book Summary: Nolan Wallach's mathematical research is remarkable in both its breadth and depth. His contributions to many fields include representation theory, harmonic analysis, algebraic geometry, combinatorics, number theory, differential equations, Riemannian geometry, ring theory, and quantum information theory.
The touchstone and unifying thread running through all his work is the idea of symmetry. This volume is a collection of invited articles that pay tribute to Wallach's ideas, and show symmetry at work in a large variety of areas. It has a … Expand. Ricci flows of Ricci flat cones. Glasgow Mathematical Journal. View 1 excerpt, cites methods. On Ricci tensor in the generalized Sasakian-space-forms.
The object of the present paper is to study the properties of generalized Sasakianspace-forms. We prove the results related to Ricci symmetric, Ricci recurrent, cyclic parallel and Codazzi type Ricci … Expand. Sphere Theorems in Geometry. In this paper, we give a survey of various sphere theorems in geometry. These include the topological sphere theorem of Berger and Klingenberg as well as the differentiable version obtained by the … Expand. View 1 excerpt, references background. Isotropic Curvature and the Ricci Flow.
In this paper, we study the Ricci flow on higher dimensional compact manifolds. We prove that nonnegative isotropic curvature is preserved by the Ricci flow in dimensions greater than or equal to … Expand.
Einstein metrics and preserved curvature conditions for the Ricci flow. Let C be a cone in the space of algebraic curvature tensors. Moreover, let M,g be a compact Einstein manifold with the property that the curvature tensor of M,g lies in the interior of the cone C … Expand. Ricci Flow and the Sphere Theorem. In , 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 … Expand.
View 6 excerpts, references methods, background and results. The proof uses the fact, also … Expand. Deforming metrics in the direction of their Ricci tensors.
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