Representação de superfícies em grupos de Lie tridimensionais

Abstract

We considered the problem of representation of immersed surfaces in three-dimensional Lie groups. We search for integrability conditions assuring the existence of a conformal immersion of a given Riemann surface in some Lie group with left-invariant metric. Such compatibility conditions are found to be a first order system, consisting of a Dirac equation with geometric potentials and an extra pair of equations relating the metric and the Hopf differential. In many cases, we proved that the harmonicity of a map, defined in an open of the sphere is a sufficient condition for the existence of a conformal minimal or constant mean curvature immersion.