New Complex Analytic Methods in the Study of Non-Orientable Minimal Surfaces in $mathbb {R}^n$ (Memoirs of the American Mathematical Society) - Softcover

Antonio Alarcon (author), Franc Forstneric (author) & Francisco J. Lopez (author)

Softcover

ISBN 10: 1470441616 ISBN 13: 9781470441616

Publisher: American Mathematical Society, 2020

Synopsis

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 $\mathbb{R}^n$.

About this title

About the Author

Antonio Alarcon, Universidad de Granada, Spain.

Franc Forstneric, University of Ljubljana, Slovenia.

Francisco J. Lopez, Universidad de Granada, Spain.

Edition details

• Publisher: American Mathematical Society
• Publication date: 2020
• Language: English
• ISBN 10: 1470441616
• ISBN 13: 9781470441616
• Binding: Paperback
• Number of pages: 77