New Complex Analytic Methods in the Study of Non-Orientable Minimal Surfaces in $ Mathbb{R}^{n}
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...
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American Mathematical Society
2020
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| ISBN: | 1470441616, 9781470441616 |
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| 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 \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 Möbius 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. |
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| AbstractList | 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 Möbius 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. |
| Author | López, Francisco J Forstnerič, Franc Alarcón, Antonio |
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| Snippet | 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... |
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| SubjectTerms | Analytic subsets of affine space Local analytic geometry [See also 13-XX and 14-XX] Several complex variables and analytic spaces |
| TableOfContents | Cover -- Title page -- Chapter 1. Introduction -- 1.1. A summary of the main results -- 1.2. Basic notions of minimal surface theory -- 1.3. Approximation and general position theorems -- 1.4. Complete non-orientable minimal surfaces with Jordan boundaries -- 1.5. Proper non-orientable minimal surfaces in domains in \Rⁿ -- Chapter 2. Preliminaries -- 2.1. Conformal structures on surfaces -- 2.2. \Igot-invariant functions and 1-forms. Spaces of functions and maps -- 2.3. Homology basis and period map -- 2.4. Conformal minimal immersions of non-orientable surfaces -- 2.5. Notation -- Chapter 3. Gluing \Igot-invariant sprays and applications -- 3.1. \Igot-invariant sprays -- 3.2. Gluing \Igot-invariant sprays on \Igot-invariant Cartan pairs -- 3.3. \Igot-invariant period dominating sprays -- 3.4. Banach manifold structure of the space \CMI_{\Igot}ⁿ(\Ncal) -- 3.5. Basic approximation results -- 3.6. The Riemann-Hilbert method for non-orientable minimal surfaces -- Chapter 4. Approximation theorems for non-orientable minimal surfaces -- 4.1. A Mergelyan approximation theorem -- 4.2. A Mergelyan theorem with fixed components -- Chapter 5. A general position theorem for non-orientable minimal surfaces -- Chapter 6. Applications -- 6.1. Proper non-orientable minimal surfaces in \Rⁿ -- 6.2. Complete non-orientable minimal surfaces with fixed components -- 6.3. Complete non-orientable minimal surfaces with Jordan boundaries -- 6.4. Proper non-orientable minimal surfaces in -convex domains -- Bibliography -- Back Cover |
| Title | New Complex Analytic Methods in the Study of Non-Orientable Minimal Surfaces in $ Mathbb{R}^{n} |
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