The Gauss map of surfaces in PSL˜2(R)

Abstract

We define a Gauss map for surfaces in the universal cover of the Lie group PSL2(R) endowed with a left-invariant Riemannian metric having a 4-dimensional isometry group. This Gauss map is not related to the Lie group structure. We prove that the Gauss map of a nowhere vertical surface of critical constant mean curvature is harmonic into the hyperbolic plane H2 and we obtain a Weierstrass-type representation formula. This extends results in H2 ×R and the Heisenberg group Nil3, and completes the proof of existence of harmonic Gauss maps for surfaces of critical constant mean curvature in any homogeneous manifold diffeomorphic to R3 with isometry group of dimension at least 4.