Spacelike surfaces of constant mean curvature ±1 in de sitter 3-space S13(1)

It is shown that spacelike surfaces of constant mean curvature +/- 1 (abbreviated as CMC +/- 1) in de Sitter 3-space S-1(3)(1) can be constructed from holomorphic curves in PSL(2; C) = SL(2; C){+/- id} via a Bryant type representation formula. This Bryant type representation formula is used to investigate an explicit one-to-one correspondence, the so-called Lawson correspondence, between spacelike CMC 1 surfaces in de Sitter 3-space S-1(3)(1) and spacelike maximal surfaces in Lorentz 3-space E-1(3). The hyperbolic Gauss map of spacelike surfaces in S-1(3)(1), which is a close analogue of the classical Gauss map, is considered. It is shown that the hyperbolic Gauss map plays all important role in the study of spacelike CMC +/- 1 surfaces ill S-1(3)(1). In particular, the relationship between the holomorphicity of the hyperbolic Gauss map and spacelike CMC +/- 1 surfaces in S-1(3)(1) is studied.