Algebraic zero mean curvature varieties in semi-riemannian manifolds

In this paper we provide a family of algebraic space-like surfaces in the three dimensional anti de Sitter space that shows that this Lorentzian manifold admits algebraic maximal examples of any order. Then, we classify all the space-like order two algebraic maximal hypersurfaces in the anti de Sitter $N$-dimensional space. Finally, we provide two families of examples of Lorentzian order two algebraic zero mean curvature in the de Sitter space.