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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, Franc Forstnerič and Francisco J. López

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

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Preliminaries
  • Chapter 3. Gluing $\Igot $-invariant sprays and applications
  • Chapter 4. Approximation theorems for non-orientable minimal surfaces
  • Chapter 5. A general position theorem for non-orientable minimal surfaces
  • Chapter 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 for any . 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 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 (see Theorem 1.4). We also give the first known example of a properly embedded non-orientable minimal surface in ; 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 with any given conformal structure (see Theorem 1.6 (i)), complete non-orientable minimal surfaces in with arbitrary conformal type whose generalized Gauss map is nondegenerate and omits hyperplanes of 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 -convex domains of (see Theorem 1.7).

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