Hypersurfaces in the non-flat Lorentzian space forms with a characteristic eigenvector field

Abstract

In a series of early papers, with the aim of knowing of the shape of a pseudo-Riemannian hypersurface satisfying a certain differential equation in the induced Laplacian, we found a remarkable family of hypersurfaces in the Lorentz-Minkowski space whose mean curvature vector is an eigenvector of the Laplacian. Actually, the last two authors showed in [8] that the equation ∆H = λH , for a real constant λ, characterizes the family of surfaces in L3 made up of the quite interesting B-scrolls and the so-called standard examples, as well as minimal surfaces. Looking at those results obtained for surfaces in L3, the following geometric question was stated in [9] for hypersurfaces in Ln+1 (n > 2): Does the equation ∆H = λH mean that both the mean and the scalar curvatures of the hypersurface are constant? We were able to give a partial solution to that problem, since we had needed to do an additional hypothesis on the degree of the minimal polynomial of the shape operator. It is worth pointing out that the additional assumption was mainly made to control the position vector field of the hypersurface into Ln+1. Now when the ambient space is a non-flat pseudoRiemannian space form, Sn+1 ν (r) or Hn+1 ν (r), then the hypersurface is of codimension two in Rn+2 ν or R ν+1 , respectively, but Sn+1 ν (r) and Hn+1 ν (r) being both totally umbilical hypersurfaces in the corresponding pseudo-Euclidean space, it seems reasonable to hope for a richer classification of hypersurfaces into those spaces by means of the equation ∆H = λH . Or even, one looks for getting a complete answer to the stated problem in non-flat ambient spaces. In this paper we give a classification of surfaces in the 3-dimensional non-flat Lorentzian space forms satisfying the equation ∆H = λH . We show that the family of such surfaces consists of minimal, totally umbilical and B-scroll surfaces. As for hypersurfaces we suppose that their shape operators have no complex eigenvalues. This condition does not seem as restrictive as one could think, in view of examples and results given in section 5. Actually, we find that family is set up by minimal, totally umbilical and so-called generalized umbilical hypersurfaces, which are nothing but a natural generalization of B-scrolls.