A Course In Minimal Surfaces

Author : Tobias H. Colding,William P. Minicozzi
Publisher : American Mathematical Soc.
Page : 330 pages
File Size : 52,5 Mb
Release : 2011
Category : Minimal surfaces
ISBN : 9780821853238

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

"Minimal surfaces date back to Euler and Lagrange and the beginning of the calculus of variations. Many of the techniques developed have played key roles in geometry and partial differential equations. Examples include monotonicity and tangent cone analysis originating in the regularity theory for minimal surfaces, estimates for nonlinear equations based on the maximum principle arising in Bernstein's classical work, and even Lebesgue's definition of the integral that he developed in his thesis on the Plateau problem for minimal surfaces. This book starts with the classical theory of minimal surfaces and ends up with current research topics. Of the various ways of approaching minimal surfaces (from complex analysis, PDE, or geometric measure theory), the authors have chosen to focus on the PDE aspects of the theory. The book also contains some of the applications of minimal surfaces to other fields including low dimensional topology, general relativity, and materials science."--Publisher's description.

Author : Tobias Holck Colding,William P. Minicozzi II
Publisher : American Mathematical Society
Page : 330 pages
File Size : 42,9 Mb
Release : 2024-01-18
Category : Mathematics
ISBN : 9781470476403

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A Course in Minimal Surfaces by Tobias Holck Colding,William P. Minicozzi II Pdf

Minimal surfaces date back to Euler and Lagrange and the beginning of the calculus of variations. Many of the techniques developed have played key roles in geometry and partial differential equations. Examples include monotonicity and tangent cone analysis originating in the regularity theory for minimal surfaces, estimates for nonlinear equations based on the maximum principle arising in Bernstein's classical work, and even Lebesgue's definition of the integral that he developed in his thesis on the Plateau problem for minimal surfaces. This book starts with the classical theory of minimal surfaces and ends up with current research topics. Of the various ways of approaching minimal surfaces (from complex analysis, PDE, or geometric measure theory), the authors have chosen to focus on the PDE aspects of the theory. The book also contains some of the applications of minimal surfaces to other fields including low dimensional topology, general relativity, and materials science. The only prerequisites needed for this book are a basic knowledge of Riemannian geometry and some familiarity with the maximum principle.

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

Author : Robert Osserman
Publisher : Springer Science & Business Media
Page : 279 pages
File Size : 40,7 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9783662034842

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Geometry V by Robert Osserman Pdf

Few people outside of mathematics are aware of the varieties of mathemat ical experience - the degree to which different mathematical subjects have different and distinctive flavors, often attractive to some mathematicians and repellant to others. The particular flavor of the subject of minimal surfaces seems to lie in a combination of the concreteness of the objects being studied, their origin and relation to the physical world, and the way they lie at the intersection of so many different parts of mathematics. In the past fifteen years a new component has been added: the availability of computer graphics to provide illustrations that are both mathematically instructive and esthetically pleas ing. During the course of the twentieth century, two major thrusts have played a seminal role in the evolution of minimal surface theory. The first is the work on the Plateau Problem, whose initial phase culminated in the solution for which Jesse Douglas was awarded one of the first two Fields Medals in 1936. (The other Fields Medal that year went to Lars V. Ahlfors for his contributions to complex analysis, including his important new insights in Nevanlinna Theory.) The second was the innovative approach to partial differential equations by Serge Bernstein, which led to the celebrated Bernstein's Theorem, stating that the only solution to the minimal surface equation over the whole plane is the trivial solution: a linear function.

Author : Robert Osserman
Publisher : Courier Corporation
Page : 232 pages
File Size : 53,6 Mb
Release : 1986
Category : Mathematics
ISBN : UOM:39015015605903

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A Survey of Minimal Surfaces by Robert Osserman Pdf

This clear and comprehensive study features 12 sections that discuss parametric and non-parametric surfaces, surfaces that minimize area, isothermal parameters, Bernstein's theorem, minimal surfaces with boundary, and many other topics. 1969 edition.

Author : W.H. III Meeks,A. Ros,H. Rosenberg
Publisher : Springer
Page : 124 pages
File Size : 48,7 Mb
Release : 2004-10-11
Category : Mathematics
ISBN : 9783540456094

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The Global Theory of Minimal Surfaces in Flat Spaces by W.H. III Meeks,A. Ros,H. Rosenberg Pdf

In the second half of the twentieth century the global theory of minimal surface in flat space had an unexpected and rapid blossoming. Some of the classical problems were solved and new classes of minimal surfaces found. Minimal surfaces are now studied from several different viewpoints using methods and techniques from analysis (real and complex), topology and geometry. In this lecture course, Meeks, Ros and Rosenberg, three of the main architects of the modern edifice, present some of the more recent methods and developments of the theory. The topics include moduli, asymptotic geometry and surfaces of constant mean curvature in the hyperbolic space.

Author : Jon T. Pitts
Publisher : Princeton University Press
Page : 337 pages
File Size : 55,7 Mb
Release : 2014-07-14
Category : Mathematics
ISBN : 9781400856459

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Existence and Regularity of Minimal Surfaces on Riemannian Manifolds. (MN-27) by Jon T. Pitts Pdf

Mathematical No/ex, 27 Originally published in 1981. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Author : Ulrich Dierkes,Stefan Hildebrandt,Friedrich Sauvigny
Publisher : Springer
Page : 692 pages
File Size : 44,8 Mb
Release : 2010-10-01
Category : Mathematics
ISBN : 3642116973

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Minimal Surfaces by Ulrich Dierkes,Stefan Hildebrandt,Friedrich Sauvigny Pdf

Minimal Surfaces is the first volume of a three volume treatise on minimal surfaces (Grundlehren Nr. 339-341). Each volume can be read and studied independently of the others. The central theme is boundary value problems for minimal surfaces. The treatise is a substantially revised and extended version of the monograph Minimal Surfaces I, II (Grundlehren Nr. 295 & 296). The first volume begins with an exposition of basic ideas of the theory of surfaces in three-dimensional Euclidean space, followed by an introduction of minimal surfaces as stationary points of area, or equivalently, as surfaces of zero mean curvature. The final definition of a minimal surface is that of a nonconstant harmonic mapping X: \Omega\to\R^3 which is conformally parametrized on \Omega\subset\R^2 and may have branch points. Thereafter the classical theory of minimal surfaces is surveyed, comprising many examples, a treatment of Björling ́s initial value problem, reflection principles, a formula of the second variation of area, the theorems of Bernstein, Heinz, Osserman, and Fujimoto. The second part of this volume begins with a survey of Plateau ́s problem and of some of its modifications. One of the main features is a new, completely elementary proof of the fact that area A and Dirichlet integral D have the same infimum in the class C(G) of admissible surfaces spanning a prescribed contour G. This leads to a new, simplified solution of the simultaneous problem of minimizing A and D in C(G), as well as to new proofs of the mapping theorems of Riemann and Korn-Lichtenstein, and to a new solution of the simultaneous Douglas problem for A and D where G consists of several closed components. Then basic facts of stable minimal surfaces are derived; this is done in the context of stable H-surfaces (i.e. of stable surfaces of prescribed mean curvature H), especially of cmc-surfaces (H = const), and leads to curvature estimates for stable, immersed cmc-surfaces and to Nitsche ́s uniqueness theorem and Tomi ́s finiteness result. In addition, a theory of unstable solutions of Plateau ́s problems is developed which is based on Courant ́s mountain pass lemma. Furthermore, Dirichlet ́s problem for nonparametric H-surfaces is solved, using the solution of Plateau ́s problem for H-surfaces and the pertinent estimates.

Author : Robert Osserman
Publisher : Unknown
Page : 288 pages
File Size : 52,9 Mb
Release : 2014-01-15
Category : Electronic
ISBN : 3662034859

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Geometry V by Robert Osserman Pdf

Author : Shoshichi Kobayashi
Publisher : Springer Nature
Page : 192 pages
File Size : 49,8 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 : William Meeks,Joaquín Pérez
Publisher : American Mathematical Soc.
Page : 195 pages
File Size : 54,8 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 : Lyndon Woodward,John Bolton
Publisher : Cambridge University Press
Page : 275 pages
File Size : 41,6 Mb
Release : 2019
Category : Mathematics
ISBN : 9781108424936

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A First Course in Differential Geometry by Lyndon Woodward,John Bolton Pdf

With detailed explanations and numerous examples, this textbook covers the differential geometry of surfaces in Euclidean space.

Author : Giusti
Publisher : Springer Science & Business Media
Page : 250 pages
File Size : 49,7 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9781468494860

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Minimal Surfaces and Functions of Bounded Variation by Giusti Pdf

The problem of finding minimal surfaces, i. e. of finding the surface of least area among those bounded by a given curve, was one of the first considered after the foundation of the calculus of variations, and is one which received a satis factory solution only in recent years. Called the problem of Plateau, after the blind physicist who did beautiful experiments with soap films and bubbles, it has resisted the efforts of many mathematicians for more than a century. It was only in the thirties that a solution was given to the problem of Plateau in 3-dimensional Euclidean space, with the papers of Douglas [DJ] and Rado [R T1, 2]. The methods of Douglas and Rado were developed and extended in 3-dimensions by several authors, but none of the results was shown to hold even for minimal hypersurfaces in higher dimension, let alone surfaces of higher dimension and codimension. It was not until thirty years later that the problem of Plateau was successfully attacked in its full generality, by several authors using measure-theoretic methods; in particular see De Giorgi [DG1, 2, 4, 5], Reifenberg [RE], Federer and Fleming [FF] and Almgren [AF1, 2]. Federer and Fleming defined a k-dimensional surface in IR" as a k-current, i. e. a continuous linear functional on k-forms. Their method is treated in full detail in the splendid book of Federer [FH 1].

Author : Anthony Tromba
Publisher : Springer Science & Business Media
Page : 194 pages
File Size : 50,8 Mb
Release : 2012-01-05
Category : Mathematics
ISBN : 9783642256202

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A Theory of Branched Minimal Surfaces by Anthony Tromba Pdf

One of the most elementary questions in mathematics is whether an area minimizing surface spanning a contour in three space is immersed or not; i.e. does its derivative have maximal rank everywhere. The purpose of this monograph is to present an elementary proof of this very fundamental and beautiful mathematical result. The exposition follows the original line of attack initiated by Jesse Douglas in his Fields medal work in 1931, namely use Dirichlet's energy as opposed to area. Remarkably, the author shows how to calculate arbitrarily high orders of derivatives of Dirichlet's energy defined on the infinite dimensional manifold of all surfaces spanning a contour, breaking new ground in the Calculus of Variations, where normally only the second derivative or variation is calculated. The monograph begins with easy examples leading to a proof in a large number of cases that can be presented in a graduate course in either manifolds or complex analysis. Thus this monograph requires only the most basic knowledge of analysis, complex analysis and topology and can therefore be read by almost anyone with a basic graduate education.

Author : Kichoon Yang
Publisher : Springer Science & Business Media
Page : 185 pages
File Size : 52,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789400910157

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Complete and Compact Minimal Surfaces by Kichoon Yang Pdf

'Et moi ..., si j'avait su comment en reveni.r, One service mathematics has rendered the je n'y serais point aile.' human race. It has put common sense back Jules Verne where it belongs. on the topmost shelf next to the dusty canister labelled 'discarded non 111e series is divergent; therefore we may be sense'. Eric T. Bell able to do something with it. O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.