Timelike surfaces of constant mean curvature ±1 in anti-de Sitter 3-space H_1~3(-1)

摘要

It is shown that timelike surfaces of constant mean curvature ± in anti-de Sitter 3-space H_1~3(-1)(-1) can be constructed from a pair of Lorentz holomorphic and Lorentz antiholomorphic null curves in PSL2R via Bryant type representation formulae. These Bryant type representation formulae are used to investigate an explicit one-to-one correspondence, the so-called Lawson–Guichard correspondence, between timelike surfaces of constant mean curvature ± 1 and timelike minimal surfaces in Minkowski 3-space E 3 1. The hyperbolic GauB map of timelike surfaces in H_1~3(-1)(-1), which is a close analogue of the classical GauB map is considered. It is discussed that the hyperbolic GauB map plays an important role in the study of timelike surfaces of constant mean curvature ± 1 in H_1~3(-1)(-1). In particular, the relationship between the Lorentz holomorphicity of the hyperbolic Gau? map and timelike surface of constant mean curvature ± 1 in H_1~3(-1)(-1) is studied.