Space-like surfaces of constant mean curvature in de Sitter 3-space S31

A Bryant type representation formula for space-like surfaces of constant mean curvature 1 (abbreviated as CMC 1) in de Sitter 3-space S31 is obtained. The formula is used to investigate a correspondence between CMC 1 space-like surfaces in S31 and maximal space-like surfaces in the Minkowski 3-space L3 . Three types of Gauss maps are discussed, and their relationships to each other are investigated. The Weierstrass type representation formula f0=ℜez0 z1 2h1+g2 ,i 2h1-g2 ,-hg dz for the maximal space-like immersion f 0 in the Minkowski 3-space L3 can be obtained as the limit surface of CMC c ( c \u3e 0) space-like surfaces in de Sitter 3-space S31 (c2) of constant positive sectional curvature c2, as the complex Lie group SL(2, C ) collapses into the abelian group C3 . A duality property of CMC 1 space-like surfaces in S31 is also obtained and studied. Some examples of CMC 1 space-like surfaces in S31 are given, and are realized in the time-like open 3-ball with an indefinite metric by the standard stereographic projection. The graphs of these surfaces are generated by MATHEMATICA.