A Survey On Classical Minimal Surface Theory PDF Download
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Author | : William Meeks |
Publisher | : American Mathematical Soc. |
Total Pages | : 195 |
Release | : 2012 |
Genre | : Mathematics |
ISBN | : 0821869124 |
Download A Survey on Classical Minimal Surface Theory Book in PDF, ePub and Kindle
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 | : Tobias Holck Colding |
Publisher | : American Mathematical Society |
Total Pages | : 330 |
Release | : 2024-01-18 |
Genre | : Mathematics |
ISBN | : 1470476401 |
Download A Course in Minimal Surfaces Book in PDF, ePub and Kindle
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 Science & Business Media |
Total Pages | : 699 |
Release | : 2010-08-16 |
Genre | : Mathematics |
ISBN | : 3642116981 |
Download Minimal Surfaces Book in PDF, ePub and Kindle
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 | : Courier Corporation |
Total Pages | : 226 |
Release | : 2013-12-10 |
Genre | : Mathematics |
ISBN | : 0486167690 |
Download A Survey of Minimal Surfaces Book in PDF, ePub and Kindle
Newly updated accessible study covers parametric and non-parametric surfaces, isothermal parameters, Bernstein’s theorem, much more, including such recent developments as new work on Plateau’s problem and on isoperimetric inequalities. Clear, comprehensive examination provides profound insights into crucial area of pure mathematics. 1986 edition. Index.
Author | : Robert Osserman |
Publisher | : |
Total Pages | : 288 |
Release | : 2014-01-15 |
Genre | : |
ISBN | : 9783662034859 |
Download Geometry V Book in PDF, ePub and Kindle
Author | : Robert Osserman |
Publisher | : Springer Science & Business Media |
Total Pages | : 279 |
Release | : 2013-03-14 |
Genre | : Mathematics |
ISBN | : 3662034840 |
Download Geometry V Book in PDF, ePub and Kindle
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 | : W.H. III Meeks |
Publisher | : Springer |
Total Pages | : 126 |
Release | : 2004-10-11 |
Genre | : Mathematics |
ISBN | : 3540456090 |
Download The Global Theory of Minimal Surfaces in Flat Spaces Book in PDF, ePub and Kindle
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 | : Antonio Alarcón |
Publisher | : Springer Nature |
Total Pages | : 430 |
Release | : 2021-03-10 |
Genre | : Mathematics |
ISBN | : 3030690563 |
Download Minimal Surfaces from a Complex Analytic Viewpoint Book in PDF, ePub and Kindle
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 Thomas Nimchek |
Publisher | : |
Total Pages | : 168 |
Release | : 2016 |
Genre | : Minimal surfaces |
ISBN | : |
Download Classical Minimal Surfaces Book in PDF, ePub and Kindle
Minimal surfaces are surfaces that minimize area locally. They have captured the imagination of mathematicians since they were first discovered in the late eighteenth century. The theory of minimal surfaces remains an active area of research. This manuscript explores classical minimal surfaces, i.e., minimal surfaces that were discovered during the eighteenth and nineteenth centuries. The minimal surface equation is derived using methods from the Calculus of Variations and Gauss' Divergence Theorem. This equation is then converted into a useful partial differential equation in both Cartesian and polar coordinates. The minimal surface equation is shown to be consistent with vanishing mean curvature. The catenoid is proven to be the only non-trivial surface of revolution that is a minimal surface (a result first proved by Muesnier in 1776) and the helicoid is proven to be the only non-trivial ruled minimal surface (a result first proved by Catalan in 1842). Scherk's surface is proved to be the only non-trivial minimal translation surface based on profile curves that exist in orthogonal planes. Separation of variables is then considered, i.e., surfaces of the form u = f(x)g(y), leading to a Cartesian expression for the helicoid. A search for other minimal surfaces based on separation of variables is undertaken. Though no new minimal surfaces are found, a method for identifying possible candidates is explored. Finally, theWeierstrass Representation Formula is introduced and used to generate the graphs of many beautiful minimal surfaces.
Author | : Antonio Alarcón |
Publisher | : American Mathematical Soc. |
Total Pages | : 77 |
Release | : 2020-05-13 |
Genre | : Education |
ISBN | : 1470441616 |
Download New Complex Analytic Methods in the Study of Non-Orientable Minimal Surfaces in Rn Book in PDF, ePub and Kindle
All the new tools mentioned above apply to non-orientable minimal surfaces endowed with a fixed choice of a conformal structure. This enables the authors to obtain significant new applications to the global theory of non-orientable minimal surfaces. In particular, they construct proper non-orientable conformal minimal surfaces in Rn with any given conformal structure, complete non-orientable minimal surfaces in Rn with arbitrary conformal type whose generalized Gauss map is nondegenerate and omits n hyperplanes of CPn−1 in general position, complete non-orientable minimal surfaces bounded by Jordan curves, and complete proper non-orientable minimal surfaces normalized by bordered surfaces in p-convex domains of Rn.