Abstract

A Bryant type representation formula for space-like surfaces of constant mean curvature 1 (abbreviated as CMC 1) in de Sitter 3-space [special characters omitted] is obtained. The formula is used to investigate a correspondence between CMC 1 space-like surfaces in [special characters omitted] and maximal space-like surfaces in the Minkowski 3-space [special characters omitted]. Three types of Gauss maps are discussed, and their relationships to each other are investigated. The Weierstrass type representation formula [special characters omitted]for the maximal space-like immersion f ₀ in the Minkowski 3-space [special characters omitted] can be obtained as the limit surface of CMC c ( c > 0) space-like surfaces in de Sitter 3-space [special characters omitted](c²) of constant positive sectional curvature c², as the complex Lie group SL(2, [special characters omitted]) collapses into the abelian group [special characters omitted]. A duality property of CMC 1 space-like surfaces in [special characters omitted] is also obtained and studied. Some examples of CMC 1 space-like surfaces in [special characters omitted] 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.