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Softcover ISBN:  9781470441616 
Product Code:  MEMO/264/1283 
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Softcover ISBN:  9781470441616 
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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 nonorientable 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 nonorientable surface to \(\mathbb{R}^n\) is a real analytic Banach manifold, obtain approximation results of RungeMergelyan type for conformal minimal immersions from nonorientable surfaces, and show general position theorems for nonorientable conformal minimal surfaces in \(\mathbb{R}^n\). The authors also give the first known example of a properly embedded nonorientable minimal surface in \(\mathbb{R}^4\); a Möbius strip.
All the new tools mentioned above apply to nonorientable 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 nonorientable minimal surfaces. In particular, they construct proper nonorientable conformal minimal surfaces in \(\mathbb{R}^n\) with any given conformal structure, complete nonorientable minimal surfaces in \(\mathbb{R}^n\) with arbitrary conformal type whose generalized Gauss map is nondegenerate and omits \(n\) hyperplanes of \(\mathbb{CP}^{n1}\) in general position, complete nonorientable minimal surfaces bounded by Jordan curves, and complete proper nonorientable minimal surfaces normalized by bordered surfaces in \(p\)convex domains of \(\mathbb{R}^n\).

Table of Contents

Chapters

1. Introduction

2. Preliminaries

3. Gluing $\mathfrak {I}$invariant sprays and applications

4. Approximation theorems for nonorientable minimal surfaces

5. A general position theorem for nonorientable minimal surfaces

6. Applications


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 nonorientable 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 nonorientable surface to \(\mathbb{R}^n\) is a real analytic Banach manifold, obtain approximation results of RungeMergelyan type for conformal minimal immersions from nonorientable surfaces, and show general position theorems for nonorientable conformal minimal surfaces in \(\mathbb{R}^n\). The authors also give the first known example of a properly embedded nonorientable minimal surface in \(\mathbb{R}^4\); a Möbius strip.
All the new tools mentioned above apply to nonorientable 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 nonorientable minimal surfaces. In particular, they construct proper nonorientable conformal minimal surfaces in \(\mathbb{R}^n\) with any given conformal structure, complete nonorientable minimal surfaces in \(\mathbb{R}^n\) with arbitrary conformal type whose generalized Gauss map is nondegenerate and omits \(n\) hyperplanes of \(\mathbb{CP}^{n1}\) in general position, complete nonorientable minimal surfaces bounded by Jordan curves, and complete proper nonorientable minimal surfaces normalized by bordered surfaces in \(p\)convex domains of \(\mathbb{R}^n\).

Chapters

1. Introduction

2. Preliminaries

3. Gluing $\mathfrak {I}$invariant sprays and applications

4. Approximation theorems for nonorientable minimal surfaces

5. A general position theorem for nonorientable minimal surfaces

6. Applications