The Gauss map of minimal surfaces in the Anti-de Sitter space

It is proved that a pair of spinors satisfying a Dirac-type equation represent surfaces immersed in Anti-de Sitter space with prescribed mean curvature. Here, we consider Anti-de Sitter space as the Lie group SU1,1SU1,1 endowed with a one-parameter family of left-invariant metrics where only one of them is bi-invariant and corresponds to the isometric embedding of Anti-de Sitter space as a quadric in R2,2R2,2. We prove that the Gauss map of a minimal surface immersed in SU1,1SU1,1 is harmonic. Conversely, we exhibit a representation of minimal surfaces in Anti-de Sitter space in terms of a given harmonic map.