Timelike surfaces of constant mean curvature ±1 in anti-de Sitter 3-space ℍ3 1(−1)

Abstract

It is shown that timelike surfaces of constant mean curvature ± in anti-de Sitter 3-space ℍ3 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 Gaus map of timelike surfaces in ℍ3 1(−1), which is a close analogue of the classical Gaus map is considered. It is discussed that the hyperbolic Gaus map plays an important role in the study of timelike surfaces of constant mean curvature ± 1 in ℍ3 1(−1). In particular, the relationship between the Lorentz holomorphicity of the hyperbolic Gaus map and timelike surface of constant mean curvature ± 1 in ℍ3 1(−1) is studied.