New examples of maximal space like surfaces in the anti-de Sitter space

Let H 1 3 be the three-dimensional anti-de Sitter space. In this paper we will construct new examples of complete maximal space like surfaces in H 1 3 . Moreover, we will prove that any complete maximal space like surface in H 1 3 with principal curvatures ±κ bounded away from zero must be isometric to the hyperbolic cylinder. Since the new examples that we have constructed have exactly two principal curvatures everywhere, we conclude that the condition on the principal curvatures on the previous result, i.e. the condition | κ ( m ) | > c > 0 , cannot be replaced by the condition | κ ( m ) | > 0 .