Timelike surfaces of constant mean curvature 1 in anti-de Sitter 3-space

摘要

It is shown that timelike surfaces of constant mean curvature 1 in anti-deSitter 3-space can be constructed from a pair of Lorentz holomorphic andLorentz antiholomorphic null curves in PSL(2,R) via Bryant type representationformulae. These formulae are used to investigate an explicit one-to-onecorrespondence, the so-called Lawson correspondence, between timelike surfacesof constant mean curvature 1 in anti-de Sitter 3-space and timelike minimalsurfaces in Minkowski 3-space. The hyperbolic Gauss map of timelike surfaces inanti-de Sitter 3-space, which is a close analogue of the classical Gauss map isconsidered. It is discussed that the hyperbolic Gauss map plays an importantrole in the study of timelike surfaces of constant mean curvature 1 in anti-deSitter 3-space. In particular, the relationship between the Lorentzholomorphicity of the hyperbolic Gauss map and timelike surfaces of constantmean curvature 1 in anti-de Sitter 3-space is studied.