  For spacelike stationary (i.e. zero mean curvature) surfaces in 4-dimensional
Lorentz space one can naturally introduce two Gauss maps and Weierstrass
representation. In this paper we investigate their global geometry
systematically. The total Gaussian curvature is related with the surface
topology as well as the indices of the so-called good singular ends by a
generalized Jorge-Meeks formula. On the other hand, as shown by a family of
counter-examples to Osserman's theorem, finite total curvature no longer
implies that Gauss maps extend to the ends. Interesting examples include the
generalization of the classical catenoids, helicoids, the Enneper surface, and
Jorge-Meeks' k-noids. Any of them could be embedded in the 4-dimensional
Lorentz space, showing a sharp contrast with the case of 3-dimensional
Euclidean space.
