SPACELIKE SURFACES OF CONSTANT MEAN CURVATURE ±1 IN DE SITTER 3-SPACE S_1~3(1)

摘要

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 an important role in the study of spacelike CMC ±1 surfaces in 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.