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New Complex Analytic Methods in the Study of Non-Orientable Minimal Surfaces in $\mathbb{R}^n$
 
Antonio Alarcón Universidad de Granada, Granada, Spain
Franc Forstnerič University of Ljubljana, Ljubljana, Slovenia
Francisco J. López Universidad de Granada, Granada, Spain
New Complex Analytic Methods in the Study of Non-Orientable Minimal Surfaces in R^n
Softcover ISBN:  978-1-4704-4161-6
Product Code:  MEMO/264/1283
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $51.00
eBook ISBN:  978-1-4704-5812-6
Product Code:  MEMO/264/1283.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $51.00
Softcover ISBN:  978-1-4704-4161-6
eBook: ISBN:  978-1-4704-5812-6
Product Code:  MEMO/264/1283.B
List Price: $170.00 $127.50
MAA Member Price: $153.00 $114.75
AMS Member Price: $102.00 $76.50
New Complex Analytic Methods in the Study of Non-Orientable Minimal Surfaces in R^n
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New Complex Analytic Methods in the Study of Non-Orientable Minimal Surfaces in $\mathbb{R}^n$
Antonio Alarcón Universidad de Granada, Granada, Spain
Franc Forstnerič University of Ljubljana, Ljubljana, Slovenia
Francisco J. López Universidad de Granada, Granada, Spain
Softcover ISBN:  978-1-4704-4161-6
Product Code:  MEMO/264/1283
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $51.00
eBook ISBN:  978-1-4704-5812-6
Product Code:  MEMO/264/1283.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $51.00
Softcover ISBN:  978-1-4704-4161-6
eBook ISBN:  978-1-4704-5812-6
Product Code:  MEMO/264/1283.B
List Price: $170.00 $127.50
MAA Member Price: $153.00 $114.75
AMS Member Price: $102.00 $76.50
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2642020; 77 pp
    MSC: 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 \(\mathbb{R}^n\).

  • Table of Contents
     
     
    • Chapters
    • 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
    • 6. Applications
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2642020; 77 pp
MSC: 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 \(\mathbb{R}^n\).

  • Chapters
  • 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
  • 6. Applications
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