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

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Abstract

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.

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Acknowledgments

We kindly thank the referee for the remarks which led to improved presentation.

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Authors and Affiliations

  1. Departamento de Geometría y Topología, Universidad de Granada, 18071, Granada, Spain

    Antonio Alarcón

  2. Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000, Ljubljana, Slovenia

    Franc Forstnerič

  3. Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000, Ljubljana, Slovenia

    Franc Forstnerič

Authors
  1. Antonio Alarcón
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  2. Franc Forstnerič
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Correspondence to Franc Forstnerič.

Additional information

A. Alarcón is supported by Vicerrectorado de Política Científica e Investigación de la Universidad de Granada, and is partially supported by MCYT-FEDER grants MTM2007-61775 and MTM2011-22547, Junta de Andalucía Grant P09-FQM-5088, and the grant PYR-2012-3 CEI BioTIC GENIL (CEB09-0010) of the MICINN CEI Program.

F. Forstnerič is supported by the program P1-0291 and the grant J1-5432, ARRS, Republic of Slovenia.

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Alarcón, A., Forstnerič, F. The Calabi–Yau problem, null curves, and Bryant surfaces. Math. Ann. 363, 913–951 (2015). https://doi.org/10.1007/s00208-015-1189-9

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