Gromov Hyperbolic Spaces and Applications to Complex Analysis

Abstract

The goal of this chapter is to explain some connections between hyperbolicity in the sense of Gromov and complex analysis/geometry. For this, we first give a short presentation of the theory of Gromov hyperbolic spaces and their boundaries. Then, we will see that the Heisenberg group can be seen as the boundary at infinity of the complex hyperbolic space. This fact will be used in Chap. 2 to give an idea of the proof of the celebrated Mostow rigidity Theorem in this setting. In the last section, we will explain why strongly pseudoconvex domains equipped with their Kobayashi distance are hyperbolic in the sense of Gromov. As an application of the general theory of Gromov hyperbolic spaces, we get a result about the extension of biholomorphic maps in this setting. Note that the case of the Gromov hyperbolicity of more general domains is discussed in Chap. 4.