The Gauss map of minimal surfaces in the anti-de Sitter space (Q627211)

In the famous paper [Trans. Am. Math. Soc. 149, 569--573 (1970; Zbl 0199.56102)], \textit{E. Ruh} and \textit{J. Wilms} proved that a submanifold in a Euclidean space has a parallel mean curvature vector field if and only if the associated Gauss map is harmonic. In particular, a surface in a 3-dimensional Euclidean space has constant mean curvature function if and only if its Gauss map is harmonic. Later on, \textit{K. Kenmotsu} [Math. Ann. 245, 89--99 (1979; Zbl 0402.53002)] introduced a modified harmonic equation in order to describe explicitely all surfaces with a given curvature function. The integral representation formula which gives the prescribed mean curvature surfaces is called the generalized Weierstrass representation and it was revised recently by \textit{B. G. Konopelchenko} and \textit{I. A. Taimanov} [J. Phys. A, Math. Gen. 29, No.~6, 1261--1265 (1996; Zbl 0911.53007)] in terms of the Dirac-type equations. In the present paper, the authors proved that a pair of spinors satisfying a Dirac-type equation represents surfaces immersed in anti-de Sitter space with prescribed mean curvature function. Then, they define the Gauss maps for a surface in \(SU_{(1,1)}^{\tau}\), and prove that, if the surface is minimal, then the Gauss map is harmonic. Here, \(SU_{(1,1)}^{\tau}\) denotes the anti-de Sitter three-space \(SU_{(1,1)}\) endowed with a one-parameter family of left-invariant metrics. Also, a representation of minimal surfaces in \(SU_{(1,1)}^{\tau}\) in terms of a given harmonic maps is given