The Gauss map of surfaces in ~PSL_2(R)

TL;DR

This paper introduces a new Gauss map for surfaces in the universal cover of PSL_2(R), proving its harmonicity for certain surfaces and providing a Weierstrass-type representation, extending known results to a broader class of homogeneous manifolds.

Contribution

It defines a novel Gauss map for surfaces in PSL_2(R) and proves its harmonicity for critical constant mean curvature surfaces, extending previous results to new geometric contexts.

Findings

Gauss map is harmonic for critical constant mean curvature surfaces

Provides a Weierstrass-type representation formula

Extends harmonic Gauss map existence to all homogeneous 3-manifolds with at least 4-dimensional isometry group

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.