Mathematische Annalen

, Volume 363, Issue 3–4, pp 913–951 | Cite as

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



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 



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


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Departamento de Geometría y TopologíaUniversidad de GranadaGranadaSpain
  2. 2.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia
  3. 3.Institute of Mathematics, Physics and MechanicsLjubljanaSlovenia

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