Minimal Surfaces From A Complex Analytic Viewpoint

Author : Antonio Alarcón,Franc Forstnerič,Francisco J. López
Publisher : Springer Nature
Page : 430 pages
File Size : 50,6 Mb
Release : 2021-03-10
Category : Mathematics
ISBN : 9783030690564

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Minimal Surfaces from a Complex Analytic Viewpoint by Antonio Alarcón,Franc Forstnerič,Francisco J. López Pdf

This monograph offers the first systematic treatment of the theory of minimal surfaces in Euclidean spaces by complex analytic methods, many of which have been developed in recent decades as part of the theory of Oka manifolds (the h-principle in complex analysis). It places particular emphasis on the study of the global theory of minimal surfaces with a given complex structure. Advanced methods of holomorphic approximation, interpolation, and homotopy classification of manifold-valued maps, along with elements of convex integration theory, are implemented for the first time in the theory of minimal surfaces. The text also presents newly developed methods for constructing minimal surfaces in minimally convex domains of Rn, based on the Riemann–Hilbert boundary value problem adapted to minimal surfaces and holomorphic null curves. These methods also provide major advances in the classical Calabi–Yau problem, yielding in particular minimal surfaces with the conformal structure of any given bordered Riemann surface. Offering new directions in the field and several challenging open problems, the primary audience of the book are researchers (including postdocs and PhD students) in differential geometry and complex analysis. Although not primarily intended as a textbook, two introductory chapters surveying background material and the classical theory of minimal surfaces also make it suitable for preparing Masters or PhD level courses.

Author : Michael A. Brilleslyper,Michael J. Dorff,Jane M. McDougall,James S. Rolf,Lisbeth E. Schaubroeck
Publisher : American Mathematical Soc.
Page : 373 pages
File Size : 45,6 Mb
Release : 2012-12-31
Category : Mathematics
ISBN : 9781614441083

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Explorations in Complex Analysis by Michael A. Brilleslyper,Michael J. Dorff,Jane M. McDougall,James S. Rolf,Lisbeth E. Schaubroeck Pdf

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.

Author : John Oprea
Publisher : American Mathematical Soc.
Page : 469 pages
File Size : 45,8 Mb
Release : 2019-02-06
Category : Electronic
ISBN : 9781470450502

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Differential Geometry and Its Applications by John Oprea Pdf

Differential Geometry and Its Applications studies the differential geometry of surfaces with the goal of helping students make the transition from the compartmentalized courses in a standard university curriculum to a type of mathematics that is a unified whole. It mixes geometry, calculus, linear algebra, differential equations, complex variables, the calculus of variations, and notions from the sciences. That mix of ideas offers students the opportunity to visualize concepts through the use of computer algebra systems such as Maple. Differential Geometry and Its Applications emphasizes that this visualization goes hand in hand with understanding the mathematics behind the computer construction. The book is rich in results and exercises that form a continuous spectrum, from those that depend on calculation to proofs that are quite abstract.

Author : Shoshichi Kobayashi
Publisher : Springer Nature
Page : 192 pages
File Size : 51,7 Mb
Release : 2019-11-13
Category : Mathematics
ISBN : 9789811517396

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Differential Geometry of Curves and Surfaces by Shoshichi Kobayashi Pdf

This book is a posthumous publication of a classic by Prof. Shoshichi Kobayashi, who taught at U.C. Berkeley for 50 years, recently translated by Eriko Shinozaki Nagumo and Makiko Sumi Tanaka. There are five chapters: 1. Plane Curves and Space Curves; 2. Local Theory of Surfaces in Space; 3. Geometry of Surfaces; 4. Gauss–Bonnet Theorem; and 5. Minimal Surfaces. Chapter 1 discusses local and global properties of planar curves and curves in space. Chapter 2 deals with local properties of surfaces in 3-dimensional Euclidean space. Two types of curvatures — the Gaussian curvature K and the mean curvature H —are introduced. The method of the moving frames, a standard technique in differential geometry, is introduced in the context of a surface in 3-dimensional Euclidean space. In Chapter 3, the Riemannian metric on a surface is introduced and properties determined only by the first fundamental form are discussed. The concept of a geodesic introduced in Chapter 2 is extensively discussed, and several examples of geodesics are presented with illustrations. Chapter 4 starts with a simple and elegant proof of Stokes’ theorem for a domain. Then the Gauss–Bonnet theorem, the major topic of this book, is discussed at great length. The theorem is a most beautiful and deep result in differential geometry. It yields a relation between the integral of the Gaussian curvature over a given oriented closed surface S and the topology of S in terms of its Euler number χ(S). Here again, many illustrations are provided to facilitate the reader’s understanding. Chapter 5, Minimal Surfaces, requires some elementary knowledge of complex analysis. However, the author retained the introductory nature of this book and focused on detailed explanations of the examples of minimal surfaces given in Chapter 2.

Author : Gary R. Jensen,Emilio Musso,Lorenzo Nicolodi
Publisher : Springer
Page : 571 pages
File Size : 52,7 Mb
Release : 2016-04-20
Category : Mathematics
ISBN : 9783319270760

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Surfaces in Classical Geometries by Gary R. Jensen,Emilio Musso,Lorenzo Nicolodi Pdf

Designed for intermediate graduate studies, this text will broaden students' core knowledge of differential geometry providing foundational material to relevant topics in classical differential geometry. The method of moving frames, a natural means for discovering and proving important results, provides the basis of treatment for topics discussed. Its application in many areas helps to connect the various geometries and to uncover many deep relationships, such as the Lawson correspondence. The nearly 300 problems and exercises range from simple applications to open problems. Exercises are embedded in the text as essential parts of the exposition. Problems are collected at the end of each chapter; solutions to select problems are given at the end of the book. Mathematica®, MatlabTM, and Xfig are used to illustrate selected concepts and results. The careful selection of results serves to show the reader how to prove the most important theorems in the subject, which may become the foundation of future progress. The book pursues significant results beyond the standard topics of an introductory differential geometry course. A sample of these results includes the Willmore functional, the classification of cyclides of Dupin, the Bonnet problem, constant mean curvature immersions, isothermic immersions, and the duality between minimal surfaces in Euclidean space and constant mean curvature surfaces in hyperbolic space. The book concludes with Lie sphere geometry and its spectacular result that all cyclides of Dupin are Lie sphere equivalent. The exposition is restricted to curves and surfaces in order to emphasize the geometric interpretation of invariants and other constructions. Working in low dimensions helps students develop a strong geometric intuition. Aspiring geometers will acquire a working knowledge of curves and surfaces in classical geometries. Students will learn the invariants of conformal geometry and how these relate to the invariants of Euclidean, spherical, and hyperbolic geometry. They will learn the fundamentals of Lie sphere geometry, which require the notion of Legendre immersions of a contact structure. Prerequisites include a completed one semester standard course on manifold theory.

Author : William Meeks,Joaquín Pérez
Publisher : American Mathematical Soc.
Page : 195 pages
File Size : 54,6 Mb
Release : 2012
Category : Mathematics
ISBN : 9780821869123

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A Survey on Classical Minimal Surface Theory by William Meeks,Joaquín Pérez Pdf

Meeks and Pérez extend their 2011 survey article "The classical theory of Minimal surfaces" in the Bulletin of the American Mathematical Society to include other recent research results. Their topics include minimal surfaces with finite topology and more than one end, limits of embedded minimal surfaces without local area or curvature bounds, conformal structure of minimal surfaces, embedded minimal surfaces of finite genus, topological aspects of minimal surfaces, and Calabi-Yau problems. There is no index. Annotation ©2013 Book News, Inc., Portland, OR (booknews.com).

Author : Henri Anciaux
Publisher : World Scientific
Page : 184 pages
File Size : 52,8 Mb
Release : 2011
Category : Mathematics
ISBN : 9789814291248

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Minimal Submanifolds in Pseudo-Riemannian Geometry by Henri Anciaux Pdf

Since the foundational work of Lagrange on the differential equation to be satisfied by a minimal surface of the Euclidean space, the theory of minimal submanifolds have undergone considerable developments, involving techniques from related areas, such as the analysis of partial differential equations and complex analysis. On the other hand, the relativity theory has led to the study of pseudo-Riemannian manifolds, which turns out to be the most general framework for the study of minimal submanifolds. However, most of the recent books on the subject still present the theory only in the Riemannian case. For the first time, this textbook provides a self-contained and accessible introduction to the subject in the general setting of pseudo-Riemannian geometry, only assuming from the reader some basic knowledge about manifold theory. Several classical results, such as the Weierstrass representation formula for minimal surfaces, and the minimizing properties of complex submanifolds, are presented in full generality without sacrificing the clarity of exposition. Finally, a number of very recent results on the subject, including the classification of equivariant minimal hypersurfaces in pseudo-Riemannian space forms and the characterization of minimal Lagrangian surfaces in some pseudo-Khler manifolds are given.

Author : Wilhelm Schlag
Publisher : American Mathematical Society
Page : 402 pages
File Size : 52,9 Mb
Release : 2014-08-06
Category : Mathematics
ISBN : 9780821898475

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A Course in Complex Analysis and Riemann Surfaces by Wilhelm Schlag Pdf

Complex analysis is a cornerstone of mathematics, making it an essential element of any area of study in graduate mathematics. Schlag's treatment of the subject emphasizes the intuitive geometric underpinnings of elementary complex analysis that naturally lead to the theory of Riemann surfaces. The book begins with an exposition of the basic theory of holomorphic functions of one complex variable. The first two chapters constitute a fairly rapid, but comprehensive course in complex analysis. The third chapter is devoted to the study of harmonic functions on the disk and the half-plane, with an emphasis on the Dirichlet problem. Starting with the fourth chapter, the theory of Riemann surfaces is developed in some detail and with complete rigor. From the beginning, the geometric aspects are emphasized and classical topics such as elliptic functions and elliptic integrals are presented as illustrations of the abstract theory. The special role of compact Riemann surfaces is explained, and their connection with algebraic equations is established. The book concludes with three chapters devoted to three major results: the Hodge decomposition theorem, the Riemann-Roch theorem, and the uniformization theorem. These chapters present the core technical apparatus of Riemann surface theory at this level. This text is intended as a detailed, yet fast-paced intermediate introduction to those parts of the theory of one complex variable that seem most useful in other areas of mathematics, including geometric group theory, dynamics, algebraic geometry, number theory, and functional analysis. More than seventy figures serve to illustrate concepts and ideas, and the many problems at the end of each chapter give the reader ample opportunity for practice and independent study.

Author : Themistocles M. Rassias
Publisher : World Scientific
Page : 350 pages
File Size : 48,7 Mb
Release : 1992
Category : Mathematics
ISBN : 9810205562

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The Problem of Plateau by Themistocles M. Rassias Pdf

This volume consists of papers written by eminent scientists from the international mathematical community, who present the latest information concerning the problem of Plateau after its classical solution by Jesse Douglas and Tibor Rad¢. The contributing papers provide insight and perspective on various problems in modern topics of Calculus of Variations, Global Differential Geometry and Global Nonlinear Analysis as related to the problem of Plateau.

Author : Daniel Huybrechts
Publisher : Springer Science & Business Media
Page : 336 pages
File Size : 49,7 Mb
Release : 2005
Category : Computers
ISBN : 3540212906

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Complex Geometry by Daniel Huybrechts Pdf

Easily accessible Includes recent developments Assumes very little knowledge of differentiable manifolds and functional analysis Particular emphasis on topics related to mirror symmetry (SUSY, Kaehler-Einstein metrics, Tian-Todorov lemma)

Author : Mark Lʹvovich Agranovskiĭ
Publisher : American Mathematical Soc.
Page : 314 pages
File Size : 40,9 Mb
Release : 2011
Category : Mathematics
ISBN : 9780821851975

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Complex Analysis and Dynamical Systems IV by Mark Lʹvovich Agranovskiĭ Pdf

The papers in this volume cover a wide variety of topics in differential geometry, general relativity, and partial differential equations. In addition, there are several articles dealing with various aspects of Lie groups and mathematics physics. Taken together, the articles provide the reader with a panorama of activity in general relativity and partial differential equations, drawn by a number of leading figures in the field. The companion volume (Contemporary Mathematics, Volume 553) is devoted to function theory and optimization.

Author : Roman E. Maeder
Publisher : Academic Press
Page : 216 pages
File Size : 46,8 Mb
Release : 2014-05-10
Category : Mathematics
ISBN : 9781483214153

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The Mathematica® Programmer by Roman E. Maeder Pdf

The Mathematica Programmer covers the fundamental programming paradigms and applications of programming languages. This book is organized into two parts encompassing 10 chapters. Part 1 begins with an overview of the programming paradigms. This part also treats abstract data types, polymorphism and message passing, object-oriented programming, and relational databases. Part 2 looks into the practical aspects of programming languages, including in lists and power series, fractal curves, and minimal surfaces. This book will prove useful to mathematicians and computer scientists.

Author : Daniel Breaz,Michael Th. Rassias
Publisher : Springer Nature
Page : 538 pages
File Size : 42,8 Mb
Release : 2020-05-12
Category : Mathematics
ISBN : 9783030401207

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Advancements in Complex Analysis by Daniel Breaz,Michael Th. Rassias Pdf

The contributions to this volume are devoted to a discussion of state-of-the-art research and treatment of problems of a wide spectrum of areas in complex analysis ranging from pure to applied and interdisciplinary mathematical research. Topics covered include: holomorphic approximation, hypercomplex analysis, special functions of complex variables, automorphic groups, zeros of the Riemann zeta function, Gaussian multiplicative chaos, non-constant frequency decompositions, minimal kernels, one-component inner functions, power moment problems, complex dynamics, biholomorphic cryptosystems, fermionic and bosonic operators. The book will appeal to graduate students and research mathematicians as well as to physicists, engineers, and scientists, whose work is related to the topics covered.

Author : Elsa Abbena,Simon Salamon,Alfred Gray
Publisher : CRC Press
Page : 872 pages
File Size : 54,8 Mb
Release : 2017-09-06
Category : Mathematics
ISBN : 9781351992206

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Modern Differential Geometry of Curves and Surfaces with Mathematica by Elsa Abbena,Simon Salamon,Alfred Gray Pdf

Presenting theory while using Mathematica in a complementary way, Modern Differential Geometry of Curves and Surfaces with Mathematica, the third edition of Alfred Gray’s famous textbook, covers how to define and compute standard geometric functions using Mathematica for constructing new curves and surfaces from existing ones. Since Gray’s death, authors Abbena and Salamon have stepped in to bring the book up to date. While maintaining Gray's intuitive approach, they reorganized the material to provide a clearer division between the text and the Mathematica code and added a Mathematica notebook as an appendix to each chapter. They also address important new topics, such as quaternions. The approach of this book is at times more computational than is usual for a book on the subject. For example, Brioshi’s formula for the Gaussian curvature in terms of the first fundamental form can be too complicated for use in hand calculations, but Mathematica handles it easily, either through computations or through graphing curvature. Another part of Mathematica that can be used effectively in differential geometry is its special function library, where nonstandard spaces of constant curvature can be defined in terms of elliptic functions and then plotted. Using the techniques described in this book, readers will understand concepts geometrically, plotting curves and surfaces on a monitor and then printing them. Containing more than 300 illustrations, the book demonstrates how to use Mathematica to plot many interesting curves and surfaces. Including as many topics of the classical differential geometry and surfaces as possible, it highlights important theorems with many examples. It includes 300 miniprograms for computing and plotting various geometric objects, alleviating the drudgery of computing things such as the curvature and torsion of a curve in space.

Author : Tobias H. Colding,William P. Minicozzi
Publisher : Courant Institute of Mathemetical Sciences
Page : 136 pages
File Size : 44,7 Mb
Release : 1999
Category : Mathematics
ISBN : STANFORD:36105021943365

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Minimal Surfaces by Tobias H. Colding,William P. Minicozzi Pdf