Constant Mean Curvature Surfaces Harmonic Maps And Integrable Systems

Author : Frederic Hélein
Publisher : Birkhäuser
ISBN 13 : 3034883307
Total Pages : 123 pages
Book Rating : 4.06/5 ( download)

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Download or read book Constant Mean Curvature Surfaces, Harmonic Maps and Integrable Systems written by Frederic Hélein and published by Birkhäuser. This book was released on 2012-12-06 with total page 123 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book intends to give an introduction to harmonic maps between a surface and a symmetric manifold and constant mean curvature surfaces as completely integrable systems. The presentation is accessible to undergraduate and graduate students in mathematics but will also be useful to researchers. It is among the first textbooks about integrable systems, their interplay with harmonic maps and the use of loop groups, and it presents the theory, for the first time, from the point of view of a differential geometer. The most important results are exposed with complete proofs (except for the last two chapters, which require a minimal knowledge from the reader). Some proofs have been completely rewritten with the objective, in particular, to clarify the relation between finite mean curvature tori, Wente tori and the loop group approach - an aspect largely neglected in the literature. The book helps the reader to access the ideas of the theory and to acquire a unified perspective of the subject.

Author : Frédéric Hélein
Publisher : Birkhauser
ISBN 13 : 9780817665760
Total Pages : 122 pages
Book Rating : 4.65/5 ( download)

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Download or read book Constant Mean Curvature Surfaces, Harmonic Maps, and Integrable Systems written by Frédéric Hélein and published by Birkhauser. This book was released on 2001-01-01 with total page 122 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Author : John C. Wood
Publisher : Springer-Verlag
ISBN 13 : 366314092X
Total Pages : 328 pages
Book Rating : 4.24/5 ( download)

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Download or read book Harmonic Maps and Integrable Systems written by John C. Wood and published by Springer-Verlag. This book was released on 2013-07-02 with total page 328 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Author : Eric Loubeau
Publisher : American Mathematical Soc.
ISBN 13 : 0821849875
Total Pages : 296 pages
Book Rating : 4.73/5 ( download)

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Download or read book Harmonic Maps and Differential Geometry written by Eric Loubeau and published by American Mathematical Soc.. This book was released on 2011 with total page 296 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of a conference held in Cagliari, Italy, from September 7-10, 2009, to celebrate John C. Wood's 60th birthday. These papers reflect the many facets of the theory of harmonic maps and its links and connections with other topics in Differential and Riemannian Geometry. Two long reports, one on constant mean curvature surfaces by F. Pedit and the other on the construction of harmonic maps by J. C. Wood, open the proceedings. These are followed by a mix of surveys on Prof. Wood's area of expertise: Lagrangian surfaces, biharmonic maps, locally conformally Kahler manifolds and the DDVV conjecture, as well as several research papers on harmonic maps. Other research papers in the volume are devoted to Willmore surfaces, Goldstein-Pedrich flows, contact pairs, prescribed Ricci curvature, conformal fibrations, the Fadeev-Hopf model, the Compact Support Principle and the curvature of surfaces.

Author : Martin A. Guest
Publisher : American Mathematical Soc.
ISBN 13 : 0821829386
Total Pages : 370 pages
Book Rating : 4.87/5 ( download)

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Download or read book Differential Geometry and Integrable Systems written by Martin A. Guest and published by American Mathematical Soc.. This book was released on 2002 with total page 370 pages. Available in PDF, EPUB and Kindle. Book excerpt: Ideas and techniques from the theory of integrable systems are playing an increasingly important role in geometry. Thanks to the development of tools from Lie theory, algebraic geometry, symplectic geometry, and topology, classical problems are investigated more systematically. New problems are also arising in mathematical physics. A major international conference was held at the University of Tokyo in July 2000. It brought together scientists in all of the areas influenced byintegrable systems. This book is the first of three collections of expository and research articles. This volume focuses on differential geometry. It is remarkable that many classical objects in surface theory and submanifold theory are described as integrable systems. Having such a description generallyreveals previously unnoticed symmetries and can lead to surprisingly explicit solutions. Surfaces of constant curvature in Euclidean space, harmonic maps from surfaces to symmetric spaces, and analogous structures on higher-dimensional manifolds are some of the examples that have broadened the horizons of differential geometry, bringing a rich supply of concrete examples into the theory of integrable systems. Many of the articles in this volume are written by prominent researchers and willserve as introductions to the topics. It is intended for graduate students and researchers interested in integrable systems and their relations to differential geometry, topology, algebraic geometry, and physics. The second volume from this conference also available from the AMS is Integrable Systems,Topology, and Physics, Volume 309 CONM/309in the Contemporary Mathematics series. The forthcoming third volume will be published by the Mathematical Society of Japan and will be available outside of Japan from the AMS in the Advanced Studies in Pure Mathematics series.

Author : James Eells
Publisher : World Scientific
ISBN 13 : 9814502928
Total Pages : 228 pages
Book Rating : 4.24/5 ( download)

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Download or read book Two Reports on Harmonic Maps written by James Eells and published by World Scientific. This book was released on 1995-03-29 with total page 228 pages. Available in PDF, EPUB and Kindle. Book excerpt: Harmonic maps between Riemannian manifolds are solutions of systems of nonlinear partial differential equations which appear in different contexts of differential geometry. They include holomorphic maps, minimal surfaces, σ-models in physics. Recently, they have become powerful tools in the study of global properties of Riemannian and Kählerian manifolds. A standard reference for this subject is a pair of Reports, published in 1978 and 1988 by James Eells and Luc Lemaire. This book presents these two reports in a single volume with a brief supplement reporting on some recent developments in the theory. It is both an introduction to the subject and a unique source of references, providing an organized exposition of results spread throughout more than 800 papers. Contents:IntroductionOperations on Vector BundlesHarmonic MapsComposition PropertiesMaps into Manifolds of Nonpositive (≤ 0) CurvatureThe Existence Theorem for Riem N ≤ 0Maps into Flat ManifoldsHarmonic Maps between SpheresHolomorphic MapsHarmonic Maps of a SurfaceHarmonic Maps between SurfacesHarmonic Maps of Manifolds with Boundary Readership: Mathematicians and mathematical physicists. keywords:Harmonic Maps;Minimal Immersions;Totally Geodesic Maps;Kaehler Manifold;(1,1)-Geodesic Map;Dilatation;Nonpositive Sectional Curvature;Holomorphic Map;Teichmueller Map;Twistor Construction “… an interesting account of the progress made in the theory of harmonic maps until the year 1988 … this master-piece work will serve as an influence and good reference in the very active subject of harmonic maps both from the points of view of theory and applications.” Mathematics Abstracts

Author : Rafael López
Publisher : Springer Science & Business Media
ISBN 13 : 3642396267
Total Pages : 296 pages
Book Rating : 4.67/5 ( download)

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Download or read book Constant Mean Curvature Surfaces with Boundary written by Rafael López and published by Springer Science & Business Media. This book was released on 2013-08-31 with total page 296 pages. Available in PDF, EPUB and Kindle. Book excerpt: The study of surfaces with constant mean curvature (CMC) is one of the main topics in classical differential geometry. Moreover, CMC surfaces are important mathematical models for the physics of interfaces in the absence of gravity, where they separate two different media or for capillary phenomena. Further, as most techniques used in the theory of CMC surfaces not only involve geometric methods but also PDE and complex analysis, the theory is also of great interest for many other mathematical fields. While minimal surfaces and CMC surfaces in general have already been treated in the literature, the present work is the first to present a comprehensive study of “compact surfaces with boundaries,” narrowing its focus to a geometric view. Basic issues include the discussion whether the symmetries of the curve inherit to the surface; the possible values of the mean curvature, area and volume; stability; the circular boundary case and the existence of the Plateau problem in the non-parametric case. The exposition provides an outlook on recent research but also a set of techniques that allows the results to be expanded to other ambient spaces. Throughout the text, numerous illustrations clarify the results and their proofs. The book is intended for graduate students and researchers in the field of differential geometry and especially theory of surfaces, including geometric analysis and geometric PDEs. It guides readers up to the state-of-the-art of the theory and introduces them to interesting open problems.

Author : Martin A. Guest
Publisher : Advanced Studies in Pure Mathe
ISBN 13 :
Total Pages : 528 pages
Book Rating : 4.91/5 ( download)

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Download or read book Surveys on Geometry and Integrable Systems written by Martin A. Guest and published by Advanced Studies in Pure Mathe. This book was released on 2008 with total page 528 pages. Available in PDF, EPUB and Kindle. Book excerpt: The articles in this volume provide a panoramic view of the role of geometry in integrable systems, firmly rooted in surface theory but currently branching out in all directions.The longer articles by Bobenko (the Bonnet problem), Dorfmeister (the generalized Weierstrass representation), Joyce (special Lagrangian 3-folds) and Terng (geometry of soliton equations) are substantial surveys of several aspects of the subject. The shorter ones indicate more briefly how the classical ideas have spread throughout differential geometry, symplectic geometry, algebraic geometry, and theoretical physics.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America

Author : Yuanlong Xin
Publisher : Springer Science & Business Media
ISBN 13 : 1461240840
Total Pages : 252 pages
Book Rating : 4.46/5 ( download)

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Download or read book Geometry of Harmonic Maps written by Yuanlong Xin and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 252 pages. Available in PDF, EPUB and Kindle. Book excerpt: Harmonic maps are solutions to a natural geometrical variational prob lem. This notion grew out of essential notions in differential geometry, such as geodesics, minimal surfaces and harmonic functions. Harmonic maps are also closely related to holomorphic maps in several complex variables, to the theory of stochastic processes, to nonlinear field theory in theoretical physics, and to the theory of liquid crystals in materials science. During the past thirty years this subject has been developed extensively. The monograph is by no means intended to give a complete description of the theory of harmonic maps. For example, the book excludes a large part of the theory of harmonic maps from 2-dimensional domains, where the methods are quite different from those discussed here. The first chapter consists of introductory material. Several equivalent definitions of harmonic maps are described, and interesting examples are presented. Various important properties and formulas are derived. Among them are Bochner-type formula for the energy density and the second varia tional formula. This chapter serves not only as a basis for the later chapters, but also as a brief introduction to the theory. Chapter 2 is devoted to the conservation law of harmonic maps. Em phasis is placed on applications of conservation law to the mono tonicity formula and Liouville-type theorems.

Author : Idrisse Khemar
Publisher : American Mathematical Soc.
ISBN 13 : 0821869256
Total Pages : 234 pages
Book Rating : 4.53/5 ( download)

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Download or read book Elliptic Integrable Systems written by Idrisse Khemar and published by American Mathematical Soc.. This book was released on 2012 with total page 234 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper, the author studies all the elliptic integrable systems, in the sense of C, that is to say, the family of all the $m$-th elliptic integrable systems associated to a $k^\prime$-symmetric space $N=G/G_0$. The author describes the geometry behind this family of integrable systems for which we know how to construct (at least locally) all the solutions. The introduction gives an overview of all the main results, as well as some related subjects and works, and some additional motivations.