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

About this Title

Antonio Alarcón, Departamento de Geometría y Topología e Instituto de Matemáticas (IEMath-GR), Universidad de Granada, Campus de Fuentenueva s/n, E–18071 Granada, Spain, Franc Forstnerič, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI–1000 Ljubljana, Slovenia, and Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI–1000 Ljubljana, Slovenia and Francisco J. López, Departamento de Geometría y Topología e Instituto de Matemáticas (IEMath-GR), Universidad de Granada, Campus de Fuentenueva s/n, E–18071 Granada, Spain

Publication: Memoirs of the American Mathematical Society
Publication Year: 2020; Volume 264, Number 1283
ISBNs: 978-1-4704-4161-6 (print); 978-1-4704-5812-6 (online)
DOI: https://doi.org/10.1090/memo/1283
Published electronically: March 18, 2020
Keywords: non-orientable surfaces, Riemann surfaces, minimal surfaces
MSC: Primary 53A10; Secondary 32B15, 32E30, 32H02

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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

Abstract

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 we develop essentially from the first principles, enable us 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 (see Theorem 1.1), obtain approximation results of Runge-Mergelyan type for conformal minimal immersions from non-orientable surfaces (see Theorem 1.2 and Corollary 1.3), and show general position theorems for non-orientable conformal minimal surfaces in $\mathbb {R}^n$ (see Theorem 1.4). We also give the first known example of a properly embedded non-orientable minimal surface in $\mathbb {R}^4$; a Möbius strip (see Example 6.1).

All our new tools mentioned above apply to non-orientable minimal surfaces endowed with a fixed choice of a conformal structure. This enables us to obtain significant new applications to the global theory of non-orientable minimal surfaces. In particular, we construct proper non-orientable conformal minimal surfaces in $\mathbb {R}^n$ with any given conformal structure (see Theorem 1.6 (i)), 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 (see Theorem 1.6 (iii)), complete non-orientable minimal surfaces bounded by Jordan curves (see Theorem 1.5), and complete proper non-orientable minimal surfaces normalized by bordered surfaces in $p$-convex domains of $\mathbb {R}^n$ (see Theorem 1.7).

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