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Harmonic Mappings In The Plane
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Book Synopsis Harmonic Mappings in the Plane by : Peter Duren
Download or read book Harmonic Mappings in the Plane written by Peter Duren and published by Cambridge University Press. This book was released on 2004-03-29 with total page 236 pages. Available in PDF, EPUB and Kindle. Book excerpt: Harmonic mappings in the plane are univalent complex-valued harmonic functions of a complex variable. Conformal mappings are a special case where the real and imaginary parts are conjugate harmonic functions, satisfying the Cauchy-Riemann equations. Harmonic mappings were studied classically by differential geometers because they provide isothermal (or conformal) parameters for minimal surfaces. More recently they have been actively investigated by complex analysts as generalizations of univalent analytic functions, or conformal mappings. Many classical results of geometric function theory extend to harmonic mappings, but basic questions remain unresolved. This book is the first comprehensive account of the theory of planar harmonic mappings, treating both the generalizations of univalent analytic functions and the connections with minimal surfaces. Essentially self-contained, the book contains background material in complex analysis and a full development of the classical theory of minimal surfaces, including the Weierstrass-Enneper representation. It is designed to introduce non-specialists to a beautiful area of complex analysis and geometry.
Book Synopsis Geometry of Harmonic Maps by : Yuanlong Xin
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.
Book Synopsis Harmonic Quasiconformal Mappings and Hyperbolic Type Metrics by : Vesna Todorčević
Download or read book Harmonic Quasiconformal Mappings and Hyperbolic Type Metrics written by Vesna Todorčević and published by Springer. This book was released on 2020-08-15 with total page 163 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book presents a research area in geometric function theory concerned with harmonic quasiconformal mappings and hyperbolic type metrics defined on planar and multidimensional domains. The classes of quasiconformal and quasiregular mappings are well established areas of study in this field as these classes are natural and fruitful generalizations of the class of analytic functions in the planar case. The book contains many concrete examples, as well as detailed proofs and explanations of motivations behind given results, gradually bringing the reader to the forefront of current research in the area. This monograph was written for a wide readership from graduate students of mathematical analysis to researchers working in this or related areas of mathematics who want to learn the tools or work on open problems listed in various parts of the book.
Book Synopsis On Harmonic Maps Into Conic Surfaces by : Jesse David Gell-Redman
Download or read book On Harmonic Maps Into Conic Surfaces written by Jesse David Gell-Redman and published by Stanford University. This book was released on 2011 with total page 133 pages. Available in PDF, EPUB and Kindle. Book excerpt: We prove the existence and uniqueness of harmonic maps in degree one homotopy classes of closed, orientable surfaces of positive genus, where the target has non-positive gauss curvature and conic points with cone angles less than $2\pi$. For a homeomorphism $w$ of such a surface, we prove existence and uniqueness of minimizers in the homotopy class of $w$ relative to the inverse images of the cone points with cone angles less than or equal to $\pi$. We show that such maps are homeomorphisms and that they depend smoothly on the target metric. For fixed geometric data, the space of minimizers in relative degree one homotopy classes is a complex manifold of (complex) dimension equal to the number of cone points with cone angles less than or equal to $\pi$. When the genus is zero, we prove the same relative minimization provided there are at least three cone points of cone angle less than or equal to $\pi$.
Book Synopsis Harmonic Maps and Differential Geometry by : Eric Loubeau
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.
Download or read book Harmonic Maps written by James Eells and published by World Scientific. This book was released on 1992 with total page 472 pages. Available in PDF, EPUB and Kindle. Book excerpt: These original research papers, written during a period of over a quarter of a century, have two main objectives. The first is to lay the foundations of the theory of harmonic maps between Riemannian Manifolds, and the second to establish various existence and regularity theorems as well as the explicit constructions of such maps.
Book Synopsis Handbook of Complex Analysis by : Reiner Kuhnau
Download or read book Handbook of Complex Analysis written by Reiner Kuhnau and published by Elsevier. This book was released on 2004-12-09 with total page 876 pages. Available in PDF, EPUB and Kindle. Book excerpt: Geometric Function Theory is that part of Complex Analysis which covers the theory of conformal and quasiconformal mappings. Beginning with the classical Riemann mapping theorem, there is a lot of existence theorems for canonical conformal mappings. On the other side there is an extensive theory of qualitative properties of conformal and quasiconformal mappings, concerning mainly a prior estimates, so called distortion theorems (including the Bieberbach conjecture with the proof of the Branges). Here a starting point was the classical Scharz lemma, and then Koebe's distortion theorem. There are several connections to mathematical physics, because of the relations to potential theory (in the plane). The Handbook of Geometric Function Theory contains also an article about constructive methods and further a Bibliography including applications eg: to electroxtatic problems, heat conduction, potential flows (in the plane). · A collection of independent survey articles in the field of GeometricFunction Theory · Existence theorems and qualitative properties of conformal and quasiconformal mappings · A bibliography, including many hints to applications in electrostatics, heat conduction, potential flows (in the plane).
Book Synopsis Explorations in Complex Analysis by : Michael A. Brilleslyper
Download or read book Explorations in Complex Analysis written by Michael A. Brilleslyper and published by American Mathematical Soc.. This book was released on 2012-12-31 with total page 373 pages. Available in PDF, EPUB and Kindle. Book excerpt: Research topics in the book include complex dynamics, minimal surfaces, fluid flows, harmonic, conformal, and polygonal mappings, and discrete complex analysis via circle packing. The nature of this book is different from many mathematics texts: the focus is on student-driven and technology-enhanced investigation. Interlaced in the reading for each chapter are examples, exercises, explorations, and projects, nearly all linked explicitly with computer applets for visualization and hands-on manipulation.
Book Synopsis Univalent Functions by : P. L. Duren
Download or read book Univalent Functions written by P. L. Duren and published by Springer Science & Business Media. This book was released on 2001-07-02 with total page 414 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Conformal Geometry written by Miao Jin and published by Springer. This book was released on 2018-04-10 with total page 314 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers an essential overview of computational conformal geometry applied to fundamental problems in specific engineering fields. It introduces readers to conformal geometry theory and discusses implementation issues from an engineering perspective. The respective chapters explore fundamental problems in specific fields of application, and detail how computational conformal geometric methods can be used to solve them in a theoretically elegant and computationally efficient way. The fields covered include computer graphics, computer vision, geometric modeling, medical imaging, and wireless sensor networks. Each chapter concludes with a summary of the material covered and suggestions for further reading, and numerous illustrations and computational algorithms complement the text. The book draws on courses given by the authors at the University of Louisiana at Lafayette, the State University of New York at Stony Brook, and Tsinghua University, and will be of interest to senior undergraduates, graduates and researchers in computer science, applied mathematics, and engineering.
Download or read book Harmonic Maps written by James Eells and published by World Scientific. This book was released on 1992-08-21 with total page 452 pages. Available in PDF, EPUB and Kindle. Book excerpt: These original research papers, written during a period of over a quarter of a century, have two main objectives. The first is to lay the foundations of the theory of harmonic maps between Riemannian Manifolds, and the second to establish various existence and regularity theorems as well as the explicit constructions of such maps. Contents:Harmonic Mappings of Riemannian Manifolds (1964)Énergie et Déformations en Géométrie Différentielle (1964)Variational Theory in Fibre Bundles (1965)Restrictions on Harmonic Maps of Surfaces (1976)The Surfaces of Delaunay (1987)Minimal Graphs (1979)On the Construction of Harmonic and Holomorphic Maps between Surfaces (1980)Deformations of Metrics and Associated Harmonic Maps (1981)A Conservation Law for Harmonic Maps (1981)Maps of Minimum Energy (1981)The Existence and Construction of Certain Harmonic Maps (1982)Harmonic Maps from Surfaces to Complex Projective Spaces (1983)Examples of Harmonic Maps from Disks to Hemispheres (1984)Variational Theory in Fibre Bundles: Examples (1983)Constructions Twistorielles des Applications Harmoniques (1983)Removable Singularities of Harmonic Maps (1984)On Equivariant Harmonic Maps (1984)Regularity of Certain Harmonic Maps (1984)Gauss Maps of Surfaces (1984)Minimal Branched Immersions into Three-Manifolds (1985)Twistorial Construction of Harmonic Maps of Surfaces into Four-Manifolds (1985)Certain Variational Principles in Riemannian Geometry (1985)Harmonic Maps and Minimal Surface Coboundaries (1987)Unstable Minimal Surface Coboundaries (1986)Harmonic Maps between Spheres and Ellipsoids (1990)On Representing Homotopy Classes by Harmonic Maps (1991) Readership: Researchers and students in differential geometry and topology and theoretical physicists. keywords:Harmonic Mapping;Energy;Holomorphic Map;First (Second) Variation of Energy;Minimal Immersion;Minimal Graph;Regularity of Maps;Removable Singularities“It is striking that the papers cut a wide swathe through mathematics, and this is a testimony to the fact that the author has influenced so many younger mathematicians, several of whom are represented here.”Mathematical Reviews
Book Synopsis Harmonic Quasiconformal Mappings and Hyperbolic Type Metrics by : Vesna Todorčević
Download or read book Harmonic Quasiconformal Mappings and Hyperbolic Type Metrics written by Vesna Todorčević and published by Springer. This book was released on 2019-07-24 with total page 163 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book presents a research area in geometric function theory concerned with harmonic quasiconformal mappings and hyperbolic type metrics defined on planar and multidimensional domains. The classes of quasiconformal and quasiregular mappings are well established areas of study in this field as these classes are natural and fruitful generalizations of the class of analytic functions in the planar case. The book contains many concrete examples, as well as detailed proofs and explanations of motivations behind given results, gradually bringing the reader to the forefront of current research in the area. This monograph was written for a wide readership from graduate students of mathematical analysis to researchers working in this or related areas of mathematics who want to learn the tools or work on open problems listed in various parts of the book.
Book Synopsis Mathematical Analysis and Computing by : R. N. Mohapatra
Download or read book Mathematical Analysis and Computing written by R. N. Mohapatra and published by Springer Nature. This book was released on 2021-05-05 with total page 661 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a collection of selected papers presented at the International Conference on Mathematical Analysis and Computing (ICMAC 2019) held at Sri Sivasubramaniya Nadar College of Engineering, Chennai, India, from 23–24 December 2019. Having found its applications in game theory, economics, and operations research, mathematical analysis plays an important role in analyzing models of physical systems and provides a sound logical base for problems stated in a qualitative manner. This book aims at disseminating recent advances in areas of mathematical analysis, soft computing, approximation and optimization through original research articles and expository survey papers. This book will be of value to research scholars, professors, and industrialists working in these areas.
Book Synopsis Linear and Complex Analysis Problem Book 3 by : Victor P. Havin
Download or read book Linear and Complex Analysis Problem Book 3 written by Victor P. Havin and published by Springer. This book was released on 2006-12-08 with total page 529 pages. Available in PDF, EPUB and Kindle. Book excerpt: The 2-volume-book is an updated, reorganized and considerably enlarged version of the previous edition of the Research Problem Book in Analysis (LNM 1043), a collection familiar to many analysts, that has sparked off much research. This new edition, created in a joint effort by a large team of analysts, is, like its predecessor, a collection of unsolved problems of modern analysis designed as informally written mini-articles, each containing not only a statement of a problem but also historical and metho- dological comments, motivation, conjectures and discussion of possible connections, of plausible approaches as well as a list of references. There are now 342 of these mini- articles, almost twice as many as in the previous edition, despite the fact that a good deal of them have been solved!
Book Synopsis Harmonic Maps Between Surfaces by : Jürgen Jost
Download or read book Harmonic Maps Between Surfaces written by Jürgen Jost and published by Springer. This book was released on 2006-12-08 with total page 143 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (PMS-48) by : Kari Astala
Download or read book Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (PMS-48) written by Kari Astala and published by Princeton University Press. This book was released on 2009-01-18 with total page 708 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book explores the most recent developments in the theory of planar quasiconformal mappings with a particular focus on the interactions with partial differential equations and nonlinear analysis. It gives a thorough and modern approach to the classical theory and presents important and compelling applications across a spectrum of mathematics: dynamical systems, singular integral operators, inverse problems, the geometry of mappings, and the calculus of variations. It also gives an account of recent advances in harmonic analysis and their applications in the geometric theory of mappings. The book explains that the existence, regularity, and singular set structures for second-order divergence-type equations--the most important class of PDEs in applications--are determined by the mathematics underpinning the geometry, structure, and dimension of fractal sets; moduli spaces of Riemann surfaces; and conformal dynamical systems. These topics are inextricably linked by the theory of quasiconformal mappings. Further, the interplay between them allows the authors to extend classical results to more general settings for wider applicability, providing new and often optimal answers to questions of existence, regularity, and geometric properties of solutions to nonlinear systems in both elliptic and degenerate elliptic settings.
