Classifying Hypersurfaces in the Lorentz-Minkowski Space with a Characteristic Eigenvector

Abstract

In a famous paper, [3], Cheng and Yau solved the Bernstein problem in the Lorentz-Minkowski space Ln+1 showing that the only entire maximal hypersurfaces are hyperplanes. Maximal and constant mean curvature (CMC) hypersurfaces play a chief role in relativity theory as it is pointed out in a series of papers by Choquet, Fischer and Marsden, [4], Stumbles, [15], and Marsden and Tipler, [13]. CMC hypersurfaces are often closely related to either an eigenvalue problem or a differential equation stemming from the Laplacian. Perhaps the most remarquable case is that concerning to vanishing constant mean curvature. Let x denote an isometric immersion of a hypersurface M in the Lorentz-Minkowski space Ln+1 and let H be the mean curvature vector field. In a recent paper, Markvorsen, [12], gives a pseudo-Riemannian version of the well-known Takahashi’s theorem showing that the coordinate functions of the immersion x are eigenfunctions of the Laplacian ∆ of M , associated to the same eigenvalue λ, if and only if M is a vanishing CMC hypersurface (λ = 0), a de Sitter space Sn 1 (r) (λ > 0) or a hyperbolic space H n(r) (λ < 0). That means that vanishing mean curvature hypersurfaces in Ln+1 are the only ones having harmonic coordinate functions. More recently, Garay and Romero, [8], ask for hypersurfaces in Ln+1 satisfying the condition ∆H = C, being C a constant vector of Ln+1 which is normal to M at every point, and show that C should vanish. As for surfaces in L3, we have shown in [7] that vanishing mean curvature surfaces are the only ones satisfying ∆H = 0, so that it seems natural to ask for the following geometric question: