The Gauss map of surfaces in ~PSL_2(R)
Differential Geometry
2013-05-08 v1
Abstract
We define a Gauss map for surfaces in the universal cover of the Lie group PSL_2(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 H^2 and we obtain a Weierstrass-type representation formula. This extends results in H^2 x R and the Heisenberg group Nil_3, and completes the proof of existence of harmonic Gauss maps for surfaces of critical constant mean curvature in any homogeneous manifold diffeomorphic to R^3 with isometry group of dimension at least 4.
Cite
@article{arxiv.1305.1491,
title = {The Gauss map of surfaces in ~PSL_2(R)},
author = {Benoit Daniel and Isabel Fernandez and Pablo Mira},
journal= {arXiv preprint arXiv:1305.1491},
year = {2013}
}