Minimal Surfaces I

Author : Ulrich Dierkes
Publisher : Springer Science & Business Media
ISBN 13 : 3662027917
Total Pages : 528 pages
Book Rating : 4.12/5 ( download)

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Download or read book Minimal Surfaces I written by Ulrich Dierkes and published by Springer Science & Business Media. This book was released on 2013-11-27 with total page 528 pages. Available in PDF, EPUB and Kindle. Book excerpt: Minimal surfaces I is an introduction to the field of minimal surfaces and apresentation of the classical theory as well as of parts of the modern development centered around boundary value problems. Part II deals with the boundary behaviour of minimal surfaces. Part I is particularly apt for students who want to enter this interesting area of analysis and differential geometry which during the last 25 years of mathematical research has been very active and productive. Surveys of various subareas will lead the student to the current frontiers of knowledge and can alsobe useful to the researcher. The lecturer can easily base courses of one or two semesters on differential geometry on Vol. 1, as many topics are worked out in great detail. Numerous computer-generated illustrations of old and new minimal surfaces are included to support intuition and imagination. Part 2 leads the reader up to the regularity theory fornonlinear elliptic boundary value problems illustrated by a particular and fascinating topic. There is no comparably comprehensive treatment of the problem of boundary regularity of minimal surfaces available in book form. This long-awaited book is a timely and welcome addition to the mathematical literature.

Author : Tobias Holck Colding
Publisher : American Mathematical Society
ISBN 13 : 1470476401
Total Pages : 330 pages
Book Rating : 4.03/5 ( download)

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Download or read book A Course in Minimal Surfaces written by Tobias Holck Colding and published by American Mathematical Society. This book was released on 2024-01-18 with total page 330 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 : Ulrich Dierkes
Publisher : Springer
ISBN 13 : 9783642116971
Total Pages : 692 pages
Book Rating : 4.73/5 ( download)

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Download or read book Minimal Surfaces written by Ulrich Dierkes and published by Springer. This book was released on 2010-10-01 with total page 692 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 : Giusti
Publisher : Springer Science & Business Media
ISBN 13 : 1468494864
Total Pages : 250 pages
Book Rating : 4.60/5 ( download)

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Download or read book Minimal Surfaces and Functions of Bounded Variation written by Giusti and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 250 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 : Ulrich Dierkes
Publisher : Springer Science & Business Media
ISBN 13 : 3642117007
Total Pages : 634 pages
Book Rating : 4.08/5 ( download)

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Download or read book Regularity of Minimal Surfaces written by Ulrich Dierkes and published by Springer Science & Business Media. This book was released on 2010-08-16 with total page 634 pages. Available in PDF, EPUB and Kindle. Book excerpt: Regularity of Minimal Surfaces begins with a survey of minimal surfaces with free boundaries. Following this, the basic results concerning the boundary behaviour of minimal surfaces and H-surfaces with fixed or free boundaries are studied. In particular, the asymptotic expansions at interior and boundary branch points are derived, leading to general Gauss-Bonnet formulas. Furthermore, gradient estimates and asymptotic expansions for minimal surfaces with only piecewise smooth boundaries are obtained. One of the main features of free boundary value problems for minimal surfaces is that, for principal reasons, it is impossible to derive a priori estimates. Therefore regularity proofs for non-minimizers have to be based on indirect reasoning using monotonicity formulas. This is followed by a long chapter discussing geometric properties of minimal and H-surfaces such as enclosure theorems and isoperimetric inequalities, leading to the discussion of obstacle problems and of Plateau ́s problem for H-surfaces in a Riemannian manifold. A natural generalization of the isoperimetric problem is the so-called thread problem, dealing with minimal surfaces whose boundary consists of a fixed arc of given length. Existence and regularity of solutions are discussed. The final chapter on branch points presents a new approach to the theorem that area minimizing solutions of Plateau ́s problem have no interior branch points.

Author : Antonio Alarcón
Publisher : Springer Nature
ISBN 13 : 3030690563
Total Pages : 430 pages
Book Rating : 4.64/5 ( download)

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Download or read book Minimal Surfaces from a Complex Analytic Viewpoint written by Antonio Alarcón and published by Springer Nature. This book was released on 2021-03-10 with total page 430 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 : U. Massari
Publisher : Elsevier
ISBN 13 : 9780080872025
Total Pages : 242 pages
Book Rating : 4.26/5 ( download)

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Download or read book Minimal Surfaces of Codimension One written by U. Massari and published by Elsevier. This book was released on 2000-04-01 with total page 242 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gives a unified presentation of different mathematical tools used to solve classical problems like Plateau's problem, Bernstein's problem, Dirichlet's problem for the Minimal Surface Equation and the Capillary problem. The fundamental idea is a quite elementary geometrical definition of codimension one surfaces. The isoperimetric property of the Euclidean balls, together with the modern theory of partial differential equations are used to solve the 19th Hilbert problem. Also included is a modern mathematical treatment of capillary problems.

Author : R. Courant
Publisher : Springer Science & Business Media
ISBN 13 : 1461299179
Total Pages : 340 pages
Book Rating : 4.72/5 ( download)

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Download or read book Dirichlet’s Principle, Conformal Mapping, and Minimal Surfaces written by R. Courant and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 340 pages. Available in PDF, EPUB and Kindle. Book excerpt: It has always been a temptation for mathematicians to present the crystallized product of their thoughts as a deductive general theory and to relegate the individual mathematical phenomenon into the role of an example. The reader who submits to the dogmatic form will be easily indoctrinated. Enlightenment, however, must come from an understanding of motives; live mathematical development springs from specific natural problems which can be easily understood, but whose solutions are difficult and demand new methods of more general significance. The present book deals with subjects of this category. It is written in a style which, as the author hopes, expresses adequately the balance and tension between the individuality of mathematical objects and the generality of mathematical methods. The author has been interested in Dirichlet's Principle and its various applications since his days as a student under David Hilbert. Plans for writing a book on these topics were revived when Jesse Douglas' work suggested to him a close connection between Dirichlet's Principle and basic problems concerning minimal sur faces. But war work and other duties intervened; even now, after much delay, the book appears in a much less polished and complete form than the author would have liked."

Author : W.H. III Meeks
Publisher : Springer
ISBN 13 : 3540456090
Total Pages : 124 pages
Book Rating : 4.94/5 ( download)

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Download or read book The Global Theory of Minimal Surfaces in Flat Spaces written by W.H. III Meeks and published by Springer. This book was released on 2004-10-11 with total page 124 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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
ISBN 13 : 1400856450
Total Pages : 337 pages
Book Rating : 4.59/5 ( download)

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Download or read book Existence and Regularity of Minimal Surfaces on Riemannian Manifolds. (MN-27) written by Jon T. Pitts and published by Princeton University Press. This book was released on 2014-07-14 with total page 337 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.