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Calabi–Bernstein Results and Parabolicity of Maximal Surfaces in Lorentzian Product Spaces

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Recent Trends in Lorentzian Geometry

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 26))

Abstract

A maximal surface in a 3-dimensional Lorentzian manifold is a space-like surface with zero mean curvature. One of the most relevant results in the context of global geometry of maximal surfaces is the well-known Calabi–Bernstein theorem, which states that the only entire maximal graphs in the 3-dimensional Lorentz–Minkowski space, \({\mathbb{R}}_{1}^{3}\), are the space-like planes. This result can also be formulated in a parametric version, stating that the only complete maximal surfaces in \({\mathbb{R}}_{1}^{3}\) are the space-like planes. In this chapter, we review about the Calabi–Bernstein theorem, summarizing some of the different extensions and generalizations of it made by several authors in recent years, and describing also some recent results obtained by the authors for maximal surfaces immersed in Lorentzian product spaces.

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Acknowledgments

This work was partially supported by MICINN and FEDER project MTM2009-10418 and Fundación Séneca project 04540/GERM/06, Spain. This research is a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Región de Murcia, Spain, by Fundación Séneca, Regional Agency for Science and Technology (Regional Plan for Science and Technology 2007–2010).

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de Córdoba, Campus Universitario de Rabanales, 14071, Córdoba, Spain

    Alma L. Albujer

  2. Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, Espinardo, 30100, Murcia, Spain

    Luis J. Alías

Authors
  1. Alma L. Albujer
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  2. Luis J. Alías
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Correspondence to Luis J. Alías .

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Editors and Affiliations

  1. , Dept of Geometry and Topology, University of Granada, Campus Fuentenueva s/n, Granada, E18071, Granada, Spain

    Miguel Sánchez

  2. Departamento de Mátematica Aplicada, Universidad de Granada, Campus de Fuentenueva, Granada, 18071, Spain

    MIguel Ortega

  3. , Dept. of Geometry and Topology, University of Granada, Campus Fuentenueva s/n, Granada, E-18071, Granada, Spain

    Alfonso Romero

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Albujer, A.L., Alías, L.J. (2012). Calabi–Bernstein Results and Parabolicity of Maximal Surfaces in Lorentzian Product Spaces. In: Sánchez, M., Ortega, M., Romero, A. (eds) Recent Trends in Lorentzian Geometry. Springer Proceedings in Mathematics & Statistics, vol 26. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4897-6_2

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