The Gauss map of surfaces in ~PSL 2 (R)

Benoit Daniel (1) , Isabel Fernandez (2) , Pablo Mira (3)

1 IECL - Institut Élie Cartan de Lorraine
2 Universidad de Sevilla = University of Seville
3 UPCT - Universidad Politécnica de Cartagena / Technical University of Cartagena

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 × 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.

Domaines: Mathématiques [math]

Benoit Daniel, Isabel Fernandez, Pablo Mira. The Gauss map of surfaces in ~PSL 2 (R). Calculus of Variations and Partial Differential Equations, 2015, 52 (3), pp.507-528. ⟨hal-01097498⟩