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New Complex Analytic Methods in the Study of Non-Orientable Minimal Surfaces in Mathbb{R}^{n}

2020 Electronic books
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Release : 2020
Genre : Electronic books
Kind : eBook
Book Rating : 126/5 ( reviews)

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Book Synopsis New Complex Analytic Methods in the Study of Non-Orientable Minimal Surfaces in Mathbb{R}^{n} by : Antonio Alarcón

Download or read book New Complex Analytic Methods in the Study of Non-Orientable Minimal Surfaces in Mathbb{R}^{n} written by Antonio Alarcón. This book was released on 2020. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this work is to adapt the complex analytic methods originating in modern Oka theory to the study of non-orientable conformal minimal surfaces in \mathbb{R}^n for any n\ge 3. These methods, which the authors develop essentially from the first principles, enable them to prove that the space of conformal minimal immersions of a given bordered non-orientable surface to \mathbb{R}^n is a real analytic Banach manifold, obtain approximation results of Runge-Mergelyan type for conformal minimal immersions from non-orientable surfaces, and show general position theorems for non-orientable co.

Minimal Surfaces from a Complex Analytic Viewpoint

2021-03-10 Mathematics
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Release : 2021-03-10
Genre : Mathematics
Kind : eBook
Book Rating : 563/5 ( reviews)

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Book Synopsis Minimal Surfaces from a Complex Analytic Viewpoint by : Antonio Alarcón

Download or read book Minimal Surfaces from a Complex Analytic Viewpoint written by Antonio Alarcón. This book was released on 2021-03-10. 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.

Global Analysis of Minimal Surfaces

2010-08-16 Mathematics
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Release : 2010-08-16
Genre : Mathematics
Kind : eBook
Book Rating : 066/5 ( reviews)

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Book Synopsis Global Analysis of Minimal Surfaces by : Ulrich Dierkes

Download or read book Global Analysis of Minimal Surfaces written by Ulrich Dierkes. This book was released on 2010-08-16. Available in PDF, EPUB and Kindle. Book excerpt: Many properties of minimal surfaces are of a global nature, and this is already true for the results treated in the first two volumes of the treatise. Part I of the present book can be viewed as an extension of these results. For instance, the first two chapters deal with existence, regularity and uniqueness theorems for minimal surfaces with partially free boundaries. Here one of the main features is the possibility of "edge-crawling" along free parts of the boundary. The third chapter deals with a priori estimates for minimal surfaces in higher dimensions and for minimizers of singular integrals related to the area functional. In particular, far reaching Bernstein theorems are derived. The second part of the book contains what one might justly call a "global theory of minimal surfaces" as envisioned by Smale. First, the Douglas problem is treated anew by using Teichmüller theory. Secondly, various index theorems for minimal theorems are derived, and their consequences for the space of solutions to Plateau ́s problem are discussed. Finally, a topological approach to minimal surfaces via Fredholm vector fields in the spirit of Smale is presented.

A Course in Minimal Surfaces

2024-01-18 Mathematics
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Release : 2024-01-18
Genre : Mathematics
Kind : eBook
Book Rating : 401/5 ( reviews)

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Book Synopsis A Course in Minimal Surfaces by : Tobias Holck Colding

Download or read book A Course in Minimal Surfaces written by Tobias Holck Colding. This book was released on 2024-01-18. 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.

Minimal Surfaces and Functions of Bounded Variation

1984-01-01 Mathematics
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Release : 1984-01-01
Genre : Mathematics
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Book Rating : 536/5 ( reviews)

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Book Synopsis Minimal Surfaces and Functions of Bounded Variation by : Giusti

Download or read book Minimal Surfaces and Functions of Bounded Variation written by Giusti. This book was released on 1984-01-01. 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].

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