Harmonic Mappings In The Plane

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Harmonic Mappings In The Plane

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Author : Peter Duren
language : en
Publisher: Cambridge University Press
Release Date : 2004-03-29

Harmonic Mappings In The Plane written by Peter Duren and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004-03-29 with Mathematics categories.

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.

On The Grid Generation Methods In Harmonic Mapping On Plane And Curved Surfaces

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Author :
language : en
Publisher:
Release Date : 1984

On The Grid Generation Methods In Harmonic Mapping On Plane And Curved Surfaces written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1984 with categories.

Harmonic Maps Between Surfaces

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Author : Jürgen Jost
language : en
Publisher: Springer
Release Date : 2006-12-08

Harmonic Maps Between Surfaces written by Jürgen Jost and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-12-08 with Mathematics categories.

Geometry Of Harmonic Maps

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Author : Yuanlong Xin
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Geometry Of Harmonic Maps written by Yuanlong Xin and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.

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.

Elliptic Partial Differential Equations And Quasiconformal Mappings In The Plane Pms 48

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Author : Kari Astala
language : en
Publisher: Princeton University Press
Release Date : 2009-01-18

Elliptic Partial Differential Equations And Quasiconformal Mappings In The Plane Pms 48 written by Kari Astala and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-01-18 with Mathematics categories.

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.

On The Existence Of Harmonic Maps From A Surface Into The Real Projective Plane A Note On Harmonic Maps Between Surfaces

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Author : J. Jost
language : en
Publisher:
Release Date : 1984

On The Existence Of Harmonic Maps From A Surface Into The Real Projective Plane A Note On Harmonic Maps Between Surfaces written by J. Jost and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1984 with categories.

Harmonic Quasiconformal Mappings And Hyperbolic Type Metrics

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Author : Vesna Todorčević
language : en
Publisher: Springer
Release Date : 2019-07-24

Harmonic Quasiconformal Mappings And Hyperbolic Type Metrics written by Vesna Todorčević and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-07-24 with Mathematics categories.

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.

N Harmonic Mappings Between Annuli

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Author : Tadeusz Iwaniec
language : en
Publisher: American Mathematical Soc.
Release Date : 2012

N Harmonic Mappings Between Annuli written by Tadeusz Iwaniec and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with Mathematics categories.

Iwaniec and Onninen (both mathematics, Syracuse U., US) address concrete questions regarding energy minimal deformations of annuli in Rn. One novelty of their approach is that they allow the mappings to slip freely along the boundaries of the domains, where it is most difficult to establish the existence, uniqueness, and invertibility properties of the extremal mappings. At the core of the matter, they say, is the underlying concept of free Lagrangians. After an introduction, they cover in turn principal radial n-harmonics, and the n-harmonic energy. There is no index. Annotation ©2012 Book News, Inc., Portland, OR (booknews.com).

On The Existence Of Harmonic Maps From A Surface Into The Real Projective Plane

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Author : Jürgen Jost
language : de
Publisher:
Release Date : 1984

On The Existence Of Harmonic Maps From A Surface Into The Real Projective Plane written by Jürgen Jost and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1984 with categories.

On Harmonic Maps Into Conic Surfaces

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Author : Jesse David Gell-Redman
language : en
Publisher: Stanford University
Release Date : 2011

On Harmonic Maps Into Conic Surfaces written by Jesse David Gell-Redman and has been published by Stanford University this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011 with categories.

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 $\pi$.