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Null Holomorphic Curves in \(\mathbb{C}^{3}\) and Applications to the Conformal Calabi-Yau Problem

  • Antonio AlarcónEmail author
  • Franc Forstnerič
Conference paper
Part of the Abel Symposia book series (ABEL, volume 10)

Abstract

In this paper we survey some recent contributions by the authors (Alarcón and Forstnerič, Math Ann 357:1049–1070, 2013; Invent Math 196:733–771, 2014; Math Ann, in press, http://link.springer.com/10.1007/s00208-015-1189-9. arXiv:1308.0903) to the theory of null holomorphic curves in the complex Euclidean space \(\mathbb{C}^{3}\), as well as their applications to null holomorphic curves in the special linear group \(SL_{2}(\mathbb{C})\), minimal surfaces in the Euclidean space \(\mathbb{R}^{3}\), and constant mean curvature one surfaces (Bryant surfaces) in the hyperbolic space \(\mathbb{H}^{3}\). The paper is an expanded version of the lecture given by the second named author at the Abel Symposium 2013 in Trondheim.

Keywords

Riemann surfaces Complex curves Null holomorphic curves Minimal surfaces Bryant surfaces Complete immersions Proper immersions 

Notes

Acknowledgements

A. Alarcón is supported by Vicerrectorado de Política Científica e Investigación de la Universidad de Granada, and is partially supported by MCYT-FEDER grant MTM2011-22547 and Junta de Andalucía Grant P09-FQM-5088. F. Forstnerič is supported by the program P1-0291 and the grant J1-5432 from ARRS, Republic of Slovenia.

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Authors and Affiliations

  1. 1.Departamento de Geometría y TopologíaUniversidad de GranadaGranadaSpain
  2. 2.Faculty of Mathematics and PhysicsInstitute of Mathematics, Physics and Mechanics, University of LjubljanaLjubljanaSlovenia

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