New Complex Analytic Methods in the Study of Non-Orientable Minimal Surfaces in $/mathbb {R}^n$
Seiten
2020
American Mathematical Society (Verlag)
978-1-4704-4161-6 (ISBN)
American Mathematical Society (Verlag)
978-1-4704-4161-6 (ISBN)
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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 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 Mobius 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 $/mathbb{R}^n$.
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 $/mathbb{R}^n$.
Antonio Alarcon, Universidad de Granada, Spain. Franc Forstneric, University of Ljubljana, Slovenia. Francisco J. Lopez, Universidad de Granada, Spain.
| Erscheinungsdatum | 01.04.2020 |
|---|---|
| Reihe/Serie | Memoirs of the American Mathematical Society |
| Verlagsort | Providence |
| Sprache | englisch |
| Maße | 178 x 254 mm |
| Gewicht | 175 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| Mathematik / Informatik ► Mathematik ► Geometrie / Topologie | |
| ISBN-10 | 1-4704-4161-6 / 1470441616 |
| ISBN-13 | 978-1-4704-4161-6 / 9781470441616 |
| Zustand | Neuware |
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