The Calabi–Yau problem, null curves, and Bryant surfaces

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• 作者：Antonio Alarcón ; Franc Forstneri?
• 关键词：53C42 ; 32H02 ; 53A10 ; 32B15
• 刊名：Mathematische Annalen
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：363
• 期：3-4
• 页码：913-951
• 全文大小：1,304 KB
• 参考文献：1.Alarcón, A., Fernández, I.: Complete minimal surfaces in \({\mathbb{R}}^3\) with a prescribed coordinate function. Differ. Geom. Appl. 29(suppl. 1), S9–S15 (2011)MATH CrossRef
  2.Alarcón, A., Fernández, I., López, F.J.: Complete minimal surfaces and harmonic functions. Comment. Math. Helv. 87, 891-04 (2012)MATH MathSciNet CrossRef
  3.Alarcón, A., Fernández, I., López, F.J.: Harmonic mappings and conformal minimal immersions of Riemann surfaces into \({\mathbb{R}}^N\) . Calc. Var. Partial Differ. Equs. 47, 227-42 (2013)MATH CrossRef
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• 作者单位：Antonio Alarcón (1)
  Franc Forstneri? (2) (3)
  
  1. Departamento de Geometría y Topología, Universidad de Granada, 18071, Granada, Spain
  2. Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000, Ljubljana, Slovenia
  3. Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000, Ljubljana, Slovenia
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1807

文摘

In this paper we prove that every bordered Riemann surface \(M\) admits a complete proper null holomorphic embedding into a ball of the complex Euclidean 3-space \(\mathbb {C}^3\). The real part of such an embedding is a complete conformal minimal immersion \(M\rightarrow \mathbb {R}^3\) with bounded image. For any such \(M\) we also construct proper null holomorphic embeddings \(M\hookrightarrow \mathbb {C}^3\) with a bounded coordinate function; these give rise to properly embedded null curves \(M\hookrightarrow SL_2(\mathbb {C})\) and to properly immersed Bryant surfaces \(M\rightarrow \mathbb {H}^3\) in the hyperbolic 3-space. In particular, we provide the first examples of proper Bryant surfaces with finite topology and of hyperbolic conformal type. The main novelty when compared to the existing results in the literature is that we work with a fixed conformal structure on \(M\). This is accomplished by introducing a conceptually new method based on complex analytic techniques. One of our main tools is an approximate solution to certain Riemann-Hilbert boundary value problems for null curves in \(\mathbb {C}^3\), developed in Sect. 3. Mathematics Subject Classification 53C42 32H02 53A10 32B15