SURFACES IN PSEUDO-RIEMANNIAN SPACE FORMS WITH ZERO MEAN CURVATURE VECTOR

We characterize a space-like surface in a pseudo-Riemannian space form with zero mean curvature vector, in terms of complex quadratic differentials on the surface as sections of a holomorphic line bundle. In addition, combining them, we have a holomorphic quartic differential. If the ambient space is S-4, then this differential is just one given in [5]. If the space is S-1(4), then the differential coincides with a holomorphic quartic differential in [6] on a Willmore surface in S-3 corresponding to the original surface through the conformal Gauss map. We define the conformal Gauss maps of surfaces in E-3 and H-3, and space-like surfaces in S-1(3), E-1(3), H-1(3) and the cone of future-directed light-like vectors of E-1(4), and have results which are analogous to those for the conformal Gauss map of a surface in S-3.