Parabolicity of maximal surfaces in Lorentzian product spaces

Abstract

In this paper we establish some parabolicity criteria for maximal surfaces immersed into a Lorentzian product space of the form M 2 × R 1 , where M 2 is a connected Riemannian surface with non-negative Gaussian curvature and M 2 × R 1 is endowed with the Lorentzian product metric , = , M − dt 2 . In particular, and as an application of our main result, we deduce that every maximal graph over a starlike domain ⊆ M is parabolic. This allows us to give an alternative proof of the non-parametric version of the Calabi-Bernstein result for entire maximal graphs in M 2 × R 1 .

Parabolicity and rigidity of spacelike hypersurfaces immersed in a Lorentzian Killing warped product

“…In this context, they obtained a new criterium to guarantee the parabolicity of complete spacelike hypersurfaces and, as application, they obtained several uniqueness results on complete maximal spacelike hypersurfaces. We also observe that, when the ambient space is a Lorentzian product space, Albujer and Alías [3], [4] obtained another very interesting rigidity results for complete maximal spacelike surfaces via the study of their parabolicity.…”

Rigidity of complete spacelike hypersurfaces with constant weighted mean curvature

“…In Li and Salavessa (2009) generalized the results of Albujer and Alías (2009) to higher dimension and codimension. Next, Albujer and Alías (2011) obtained some parabolicity criteria for maximal surfaces immersed into a Lorentzian product space R 1 × P 2 , where P 2 is supposed to have nonnegative Gaussian curvature. As an application of their main result, they deduced that every maximal graph over a starlike domain ⊂ P 2 is parabolic.…”

Entire spacelike $H$-graphs in Lorentzian product spaces

“…In [5], the first author jointly with Albujer and Camargo established uniqueness results concerning complete spacelike hypersurfaces with constant mean curvature immersed in −R × H n . Next, Albujer and Alías [4] obtained some parabolicity criteria for maximal surfaces immersed into a Lorentzian product space −R × M 2 , where M 2 is supposed to have nonnegative Gaussian curvature. As an application of their main result, they deduced that every maximal graph over a starlike domain Ω ⊂ M 2 is parabolic.…”