Screw motion surfaces in H 2× R and S 2× R

In this paper we study the geometry of constant mean curvature H-screw motion surfaces in H 2 R and S 2 R. We compute the Abresch-Rosenberg holomorphic quadratic differential Q. For instance, if the ambient space is H 2 R, we derive that, given , if 0 < 4H 2 < 1, then there exists a complete H-screw motion surface with pitch immersed in H 2 R, such that Q = 0. An analogous result holds if the ambient space is S 2 R. We deduce a general non-parametric formula for the mean curvature H(). When H is constant, we find a first integral and use it to get an explicit two parameter family of complete, embedded, simply connected, minimal screw motion surfaces in H 2 R with pitch . If = 1, each such surface has Gaussian curvature K -1. We deduce that any two isometric screw motion minimal immersions in H 2 R or S 2 R are associate, i.e., the absolute values of their Hopf functions are the same.