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For spacelike stationary (i.e. zero mean curvature) surfaces in 4-dimensionalLorentz space one can naturally introduce two Gauss maps and Weierstrassrepresentation. In this paper we investigate their global geometrysystematically. The total Gaussian curvature is related with the surfacetopology as well as the indices of the so-called good singular ends by ageneralized Jorge-Meeks formula. On the other hand, as shown by a family ofcounter-examples to Osserman's theorem, finite total curvature no longerimplies that Gauss maps extend to the ends. Interesting examples include thegeneralization of the classical catenoids, helicoids, the Enneper surface, andJorge-Meeks' k-noids. Any of them could be embedded in the 4-dimensionalLorentz space, showing a sharp contrast with the case of 3-dimensionalEuclidean space.

Global geometry and topology of spacelike stationary surfaces in the 4-dimensional Lorentz space