A aplicação de Gauss de superfícies no espaço de Heisenberg

Abstract

In this report, we study minimal surfaces of the tridimensional Heisenberg group, as well as their Gauss maps. We begin with a short presentation of the geometry of the Heisenberg group. Then, we show that, in this space: the only surfaces with constant Gauss map are the vertical planes; there are no totally umbilical surfaces nor compact minimal surfaces; every minimal surface is, necessarily, stable. We also show that the only vertical minimal surfaces are vertical planes. Finally, we present a classification of the surfaces with Gauss map of constant rank, equal to zero or one.