Surfaces in pseudo-Riemannian space forms with zero mean curvature vector

Abstract

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 S⁴, then this differential is just one given in [5]. If the space is S₁⁴, then the differential coincides with a holomorphic quartic differential in [6] on a Willmore surface in S³ corresponding to the original surface through the conformal Gauss map. We define the conformal Gauss maps of surfaces in E³ and H³, and space-like surfaces in S₁³, E₁³, H₁³ and the cone of future-directed light-like vectors of E₁⁴, and have results which are analogous to those for the conformal Gauss map of a surface in S³.