Mathematical Notes

, Volume 105, Issue 1–2, pp 280–284 | Cite as

An Extension of Calabi’s Correspondence between the Solutions of Two Bernstein Problems to More General Elliptic Nonlinear Equations

  • José A. S. PelegrínEmail author
  • Alfonso RomeroEmail author
  • Rafael. M. RubioEmail author


A new correspondence between the solutions of theminimal surface equation in a certain 3-dimensional Riemannian warped product and the solutions of the maximal surface equation in a 3-dimensional standard static space-time is given. This widely extends the classical duality between minimal graphs in 3-dimensional Euclidean space and maximal graphs in 3-dimensional Lorentz–Minkowski space-time. We highlight the fact that this correspondence can be restricted to the respective classes of entire solutions. As an application, a Calabi–Bernstein-type result for certain static standard space-times is proved.


minimal surface equation maximal surface equation Riemannian warped product manifold standard static space-time 


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Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Departamento de Geometría y TopologíaUniversidad de GranadaGranadaSpain
  2. 2.Departamento de Matemáticas, Campus de RabanalesUniversidad de CórdobaCórdobaSpain

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