A Aplicação de Gauss de superfícies imersas no grupo de Heisenberg

Abstract

In this dissertation, we investigate the geometry of the Heisenberg group (H_3) and present a generalization of the normal Gauss map for oriented surfaces immersed in three-dimensional Lie groups. Our goal is to analyze the geometric properties and implications of this construction, with emphasis on the particular case where the ambient space is the Heisenberg group. Additionally, we explore the Gauus map in the context of conformal immersions in H_3, using key tools from Complex Analysis to support this study. Among the main results presented, we highlight in the first part, those obtained by Christiam Figueroa [9], who demonstrated that vertical planes are the only connected surfaces in H3 with constant Gauss map, and that there are no compact minimal surfaces in H_3. In the second part, by considering conformal immersions in H_3, we obtain interesting geometric interpretations of the Gauss map. The theoretical foundation for this study is based on the article entitled The Gauss Map of Minimal Surfaces in the Heisenberg Group, by Daniel Benôit [2].