Aplicação de Gauss hiperbólica e superfícies CMC em H2xR

Abstract

In this work we studied geometric properties of surfaces with regular vertical projection and constant mean curvature in H^2xR. For the special case in which the constant mean curvature is 1/2, a certain map is constructed on the hyperbolic plane H^2, called Hyperbolic Gauss Map and the harmonicity of it is subsequently obtained. Another key point is that under certain conditions imposed on the surface, it is always possible from a given harmonic map G to recover a mean curvature surface H=1/2, such that G is its Hyperbolic Gauss Map and whose parameterization is given in terms of G. These results were obtained by Isabel Fernández and Pablo Mira in "Harmonic Maps and Constant Mean Curvature Surfaces in H^2xR". The demonstrations of these results make use of conformal parameters and the use of techniques known in the theory of constant mean curvature surfaces. This allows you to find initial conditions for recovering a surface from a system of partial differential equations.