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On the characterization of minimal surfaces with finite total curvature in \({\mathbb {H}}^2\times {\mathbb {R}}\) and \(\widetilde{\mathrm{PSL}}_2 ({\mathbb {R}})\)

  • Laurent HauswirthEmail author
  • Ana Menezes
  • Magdalena Rodríguez
Article
  • 41 Downloads

Abstract

It is known that a complete immersed minimal surface with finite total curvature in \({\mathbb {H}}^2\times {\mathbb {R}}\) is proper, has finite topology and each one of its ends is asymptotic to a geodesic polygon at infinity (Hauswirth and Rosenberg in Mat Contemp 31:65–80, 2006; Hauswirth et al. in Adv Math 274:199–240, 2015). In this paper we prove that these three properties characterize complete immersed minimal surfaces with finite total curvature in \({\mathbb {H}}^2\times {\mathbb {R}}\). As corollaries of this theorem we obtain characterizations for minimal Scherk-type graphs and horizontal catenoids in \({\mathbb {H}}^2\times {\mathbb {R}}\). We also prove that if a properly immersed minimal surface in \(\widetilde{\mathrm{PSL}}_2({\mathbb {R}},\tau )\) has finite topology and each one of its ends is asymptotic to a geodesic polygon at infinity, then it must have finite total curvature.

Mathematics Subject Classification

53A10 53C42 

Notes

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Laurent Hauswirth
    • 1
    Email author
  • Ana Menezes
    • 2
  • Magdalena Rodríguez
    • 3
  1. 1.Département de MathématiquesUniversité de Marne-la-ValléeMarne-la-Vallée Cedex 2France
  2. 2.Mathematics DepartmentPrinceton UniversityPrincetonUSA
  3. 3.Departamento de Geometría y TopologíaUniversidad de GranadaGranadaSpain

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