New Complex Analytic Methods in the Study of Non-orientable Minimal Surfaces in Mathbb Rn (Memoirs of the American Mathematical Society, Band 264) Forstneric, Franc, J. Lopez Francisco and Antonio Alarcon

Forstneric, Franc, J. Lopez Francisco and Antonio Alarcon

ISBN 10: 1470441616 / ISBN 13: 9781470441616

Editore: American Mathematical Society, 2020

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New Complex Analytic Methods in the Study of Non-orientable Minimal Surfaces in Mathbb Rn (Memoirs of the American Mathematical Society, Band 264) In englischer Sprache. pages.

Codice articolo BN342981

Dati bibliografici

• Titolo: New Complex Analytic Methods in the Study of Non-orientable Minimal Surfaces in Mathbb Rn (Memoirs of the American Mathematical Society, Band 264) Forstneric, Franc, J. Lopez Francisco and Antonio Alarcon
• Editore: American Mathematical Society
• Data di pubblicazione: 2020
• Lingua: Inglese
• ISBN 10: 1470441616
• ISBN 13: 9781470441616
• Rilegatura: Brossura
• Edizione: 2020.
• Condizione: gut
• Peso: 2000 grammi

Descrizione articolo

Riassunto

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 $\mathbb{R}^n$.