Constant Mean Curvature Surfaces Harmonic Maps And Integrable Systems

Author : Frederic Hélein
Publisher : Birkhäuser
Page : 122 pages
File Size : 46,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034883306

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Constant Mean Curvature Surfaces, Harmonic Maps and Integrable Systems by Frederic Hélein Pdf

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
Page : 122 pages
File Size : 55,9 Mb
Release : 2001-01-01
Category : Mathematics
ISBN : 0817665765

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Constant Mean Curvature Surfaces, Harmonic Maps, and Integrable Systems by Frédéric Hélein Pdf

Author : John C. Wood
Publisher : Springer-Verlag
Page : 328 pages
File Size : 41,5 Mb
Release : 2013-07-02
Category : Mathematics
ISBN : 9783663140924

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Harmonic Maps and Integrable Systems by John C. Wood Pdf

Author : Eric Loubeau,Stefano Montaldo
Publisher : American Mathematical Soc.
Page : 296 pages
File Size : 40,6 Mb
Release : 2011
Category : Geometry, Differential
ISBN : 9780821849873

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Harmonic Maps and Differential Geometry by Eric Loubeau,Stefano Montaldo Pdf

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,Reiko Miyaoka,Yoshihiro Ohnita
Publisher : American Mathematical Soc.
Page : 349 pages
File Size : 47,7 Mb
Release : 2002
Category : Mathematics
ISBN : 9780821829387

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Differential Geometry and Integrable Systems by Martin A. Guest,Reiko Miyaoka,Yoshihiro Ohnita Pdf

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 by integrable 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 generally reveals 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 will serve 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"" in 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,Luc Lemaire
Publisher : World Scientific
Page : 228 pages
File Size : 44,7 Mb
Release : 1995-03-29
Category : Mathematics
ISBN : 9789814502924

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Two Reports on Harmonic Maps by James Eells,Luc Lemaire Pdf

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 : James Eells,Andrea Ratto
Publisher : Princeton University Press
Page : 238 pages
File Size : 41,6 Mb
Release : 1993
Category : Mathematics
ISBN : 069110249X

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Harmonic Maps and Minimal Immersions with Symmetries by James Eells,Andrea Ratto Pdf

The aim of this book is to study harmonic maps, minimal and parallel mean curvature immersions in the presence of symmetry. In several instances, the latter permits reduction of the original elliptic variational problem to the qualitative study of certain ordinary differential equations: the authors' primary objective is to provide representative examples to illustrate these reduction methods and their associated analysis with geometric and topological applications. The material covered by the book displays a solid interplay involving geometry, analysis and topology: in particular, it includes a basic presentation of 1-cohomogeneous equivariant differential geometry and of the theory of harmonic maps between spheres.

Author : Martin A. Guest,Reiko Miyaoka,Yoshihiro Ohnita
Publisher : Advanced Studies in Pure Mathe
Page : 528 pages
File Size : 43,7 Mb
Release : 2008
Category : Mathematics
ISBN : UOM:39015075648991

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Surveys on Geometry and Integrable Systems by Martin A. Guest,Reiko Miyaoka,Yoshihiro Ohnita Pdf

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
Page : 252 pages
File Size : 48,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461240846

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Geometry of Harmonic Maps by Yuanlong Xin Pdf

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 : Rafael López
Publisher : Springer Science & Business Media
Page : 296 pages
File Size : 49,5 Mb
Release : 2013-08-31
Category : Mathematics
ISBN : 9783642396267

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Constant Mean Curvature Surfaces with Boundary by Rafael López Pdf

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 : Idrisse Khemar
Publisher : American Mathematical Soc.
Page : 217 pages
File Size : 43,7 Mb
Release : 2012
Category : Mathematics
ISBN : 9780821869253

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Elliptic Integrable Systems by Idrisse Khemar Pdf

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.

Author : Christopher Kum Anand,Paul Baird,John Colin Wood,Eric Loubeau
Publisher : CRC Press
Page : 332 pages
File Size : 48,9 Mb
Release : 1999-10-13
Category : Mathematics
ISBN : 1584880325

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Harmonic Morphisms, Harmonic Maps and Related Topics by Christopher Kum Anand,Paul Baird,John Colin Wood,Eric Loubeau Pdf

The subject of harmonic morphisms is relatively new but has attracted a huge worldwide following. Mathematicians, young researchers and distinguished experts came from all corners of the globe to the City of Brest - site of the first, international conference devoted to the fledgling but dynamic field of harmonic morphisms. Harmonic Morphisms, Harmonic Maps, and Related Topics reports the proceedings of that conference, forms the first work primarily devoted to harmonic morphisms, bringing together contributions from the founders of the subject, leading specialists, and experts in other related fields. Starting with "The Beginnings of Harmonic Morphisms," which provides the essential background, the first section includes papers on the stability of harmonic morphisms, global properties, harmonic polynomial morphisms, Bochner technique, f-structures, symplectic harmonic morphisms, and discrete harmonic morphisms. The second section addresses the wider domain of harmonic maps and contains some of the most recent results on harmonic maps and surfaces. The final section highlights the rapidly developing subject of constant mean curvature surfaces. Harmonic Morphisms, Harmonic Maps, and Related Topics offers a coherent, balanced account of this fast-growing subject that furnishes a vital reference for anyone working in the field.

Author : Yuan-Jen Chiang
Publisher : Springer Science & Business Media
Page : 418 pages
File Size : 41,8 Mb
Release : 2013-06-18
Category : Mathematics
ISBN : 9783034805346

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Developments of Harmonic Maps, Wave Maps and Yang-Mills Fields into Biharmonic Maps, Biwave Maps and Bi-Yang-Mills Fields by Yuan-Jen Chiang Pdf

Harmonic maps between Riemannian manifolds were first established by James Eells and Joseph H. Sampson in 1964. Wave maps are harmonic maps on Minkowski spaces and have been studied since the 1990s. Yang-Mills fields, the critical points of Yang-Mills functionals of connections whose curvature tensors are harmonic, were explored by a few physicists in the 1950s, and biharmonic maps (generalizing harmonic maps) were introduced by Guoying Jiang in 1986. The book presents an overview of the important developments made in these fields since they first came up. Furthermore, it introduces biwave maps (generalizing wave maps) which were first studied by the author in 2009, and bi-Yang-Mills fields (generalizing Yang-Mills fields) first investigated by Toshiyuki Ichiyama, Jun-Ichi Inoguchi and Hajime Urakawa in 2008. Other topics discussed are exponential harmonic maps, exponential wave maps and exponential Yang-Mills fields.

Author : Chaohao Gu,Anning Hu,Zixiang Zhou
Publisher : Springer Science & Business Media
Page : 317 pages
File Size : 51,5 Mb
Release : 2006-07-09
Category : Science
ISBN : 9781402030888

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Darboux Transformations in Integrable Systems by Chaohao Gu,Anning Hu,Zixiang Zhou Pdf

The Darboux transformation approach is one of the most effective methods for constructing explicit solutions of partial differential equations which are called integrable systems and play important roles in mechanics, physics and differential geometry. This book presents the Darboux transformations in matrix form and provides purely algebraic algorithms for constructing the explicit solutions. A basis for using symbolic computations to obtain the explicit exact solutions for many integrable systems is established. Moreover, the behavior of simple and multi-solutions, even in multi-dimensional cases, can be elucidated clearly. The method covers a series of important equations such as various kinds of AKNS systems in R1+n, harmonic maps from 2-dimensional manifolds, self-dual Yang-Mills fields and the generalizations to higher dimensional case, theory of line congruences in three dimensions or higher dimensional space etc. All these cases are explained in detail. This book contains many results that were obtained by the authors in the past few years. Audience: The book has been written for specialists, teachers and graduate students (or undergraduate students of higher grade) in mathematics and physics.

Author : Áurea Casinhas Quintino
Publisher : Unknown
Page : 128 pages
File Size : 52,6 Mb
Release : 2021-03
Category : Electronic
ISBN : 1108885470

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Constrained Willmore Surfaces by Áurea Casinhas Quintino Pdf

"This work is dedicated to the study of the Mèobius invariant class of constrained Willmore surfaces and its symmetries. Characterized by the perturbed harmonicity of the mean curvature sphere congruence, a generalization of the well-developed integrable systems theory of harmonic maps emerges. The starting point is a zero-curvature characterization, due to Burstall-Calderbank, which we derive from the underlying variational problem. Constrained Willmore surfaces come equipped with a family of flat metric connections. We then define a spectral deformation, by the action of a loop of flat metric connections; Bèacklund transformations, defined by the application of a version of the Terng-Uhlenbeck dressing action by simple factors; and, in 4-space, Darboux transformations, based on the solution of a Riccati equation, generalizing the transformation of Willmore surfaces presented in the quaternionic setting by Burstall-Ferus-Leschke-Pedit-Pinkall. We establish a permutability between spectral deformation and Bèacklund transformation and prove that non-trivial Darboux transformation of constrained Willmore surfaces in 4-space can be obtained as a particular case of Bèacklund transformation. All these transformations corresponding to the zero Lagrange multiplier preserve the class of Willmore surfaces. We verify that both spectral deformation and Bèacklund transformation preserve the class of constrained Willmore surfaces admitting a conserved quantity, defining, in particular, transformations within the class of constant mean curvature surfaces in 3-dimensional space-forms, with, furthermore, preservation of both the space-form and the mean curvature, in the latter case. Constrained Willmore transformation proves to be unifying to the rich transformation theory of CMC surfaces in 3-space"--