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Softcover ISBN: | 978-1-4704-4161-6 |
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Softcover ISBN: | 978-1-4704-4161-6 |
Product Code: | MEMO/264/1283 |
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Product Code: | MEMO/264/1283.E |
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Softcover ISBN: | 978-1-4704-4161-6 |
eBook ISBN: | 978-1-4704-5812-6 |
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 264; 2020; 77 ppMSC: Primary 53; Secondary 32
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 conformal minimal surfaces in \(\mathbb{R}^n\). The authors also give the first known example of a properly embedded non-orientable minimal surface in \(\mathbb{R}^4\); a Möbius strip.
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 \(\mathbb{R}^n\) with any given conformal structure, complete non-orientable minimal surfaces in \(\mathbb{R}^n\) with arbitrary conformal type whose generalized Gauss map is nondegenerate and omits \(n\) hyperplanes of \(\mathbb{CP}^{n-1}\) 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 \(\mathbb{R}^n\).
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Table of Contents
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Chapters
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1. Introduction
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2. Preliminaries
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3. Gluing $\mathfrak {I}$-invariant sprays and applications
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4. Approximation theorems for non-orientable minimal surfaces
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5. A general position theorem for non-orientable minimal surfaces
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6. Applications
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Additional Material
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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 conformal minimal surfaces in \(\mathbb{R}^n\). The authors also give the first known example of a properly embedded non-orientable minimal surface in \(\mathbb{R}^4\); a Möbius strip.
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 \(\mathbb{R}^n\) with any given conformal structure, complete non-orientable minimal surfaces in \(\mathbb{R}^n\) with arbitrary conformal type whose generalized Gauss map is nondegenerate and omits \(n\) hyperplanes of \(\mathbb{CP}^{n-1}\) 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 \(\mathbb{R}^n\).
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Chapters
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1. Introduction
-
2. Preliminaries
-
3. Gluing $\mathfrak {I}$-invariant sprays and applications
-
4. Approximation theorems for non-orientable minimal surfaces
-
5. A general position theorem for non-orientable minimal surfaces
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6. Applications