New complex analytic methods in the study of non-orientable minimal surfaces in Rn
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 immersio...
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| Sprache: | eng |
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American Mathematical Society
March 2020
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| Schriftenreihe: | Memoirs of the American Mathematical Society
volume 264, number 1283 (March 2020) |
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| Online Zugang: | Inhaltsverzeichnis |
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Tag hinzufügen | Zusammenfassung: | 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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| Beschreibung: | Literaturverzeichnis: Seite 73-77 "March 2020, volume 264, number 1283 (sixth of 6 numbers)" |
| Beschreibung: | vi, 77 Seiten Illustrationen |
| ISBN: | 9781470441616 978-1-4704-4161-6 |