Abstract
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.
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References
E. Calabi, “Examples of Bernstein problems for some nonlinear equations,” Proc. Sympos. Pure Math. 15, 223–230 (1970).
L. J. Alías and B. Palmer, “A duality result between the minimal surface equation and the maximal surface equation,” An. Acad. Bras. Cienc. 73, 161–164 (2001).
A. L. Albujer and L. J. Alías, “Calabi–Bernstein results and parabolicity of maximal surfaces in Lorentzian product spaces,” Recend trends in Lorentzian geometry, 49–85, Springer Proc. Math. Stat. 26, 2013.
O. Kobayashi, “Maximal surfaces in the 3–dimensional Lorentz–Minkowski space L3,” Tokyo J. Math. 6, 297–309 (1983).
L. V. McNertey, One–parameter families of surfaces with constant curvature in Lorentz 3–space Ph. D. thesis, Brown University, USA, 1980.
F. J. M. Estudillo and A. Romero, “Generalized maximal surfaces in Lorentz–Minkowski space L3,” Math. Proc. Cambridge 111, 515–524 (1992).
H. Arau´ jo and M. L. Leite, “How many maximal surfaces do correspond to one minimal surface?,” Math. Proc. Cambridge Philos. Soc. 146, 165–175 (2009).
A. L. Albujer and L. J. Alías, “Calabi–Bernstein results for maximal surfaces in Lorentzian product spaces,” J. Geom. Phys. 59, 620–631 (2009).
H. Lee, “Extensions of the duality between minimal surfaces and maximal surfaces,” Geom. Dedicata 151, 373–386 (2011).
H. Lee, “Maximal surfaces in Lorentzian Heisenberg space,” Differential Geom. Appl. 29, 73–84 (2011).
H. Lee, “Minimal surface systems, maximal surface systems and special Lagrangian equations,” Trans. Amer. Math. Soc. 365, 3775–3797 (2013).
B. O’Neill, Semi–Riemannian Geometry with applications to Relativity (Academic Press, 1983).
A. Romero and R. M. Rubio, “Bernstein–type Theorems in aRiemannianManifold with an Irrotational Killing Vector Field,” Mediterr. J. Math. 13, 1285–1290 (2016).
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Pelegrín, J.A., Romero, A. & Rubio, R.M. An Extension of Calabi’s Correspondence between the Solutions of Two Bernstein Problems to More General Elliptic Nonlinear Equations. Math Notes 105, 280–284 (2019). https://doi.org/10.1134/S0001434619010309
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DOI: https://doi.org/10.1134/S0001434619010309