Parole chiave

• horocycle
• holomorphic self-map
• horosphere
• Gromov compactification

Abstract A horocycle in the unit disk of the complex plane is a euclidean disk which is internally tangent to a point $p$ of the boundary of the disk. Horocycles are limits of Poincaré balls as the center moves towards the point $p$ while the radius grows suitably. The classical Julia lemma is a boundary version of the Schwarz lemma which shows that horocycles are useful to understand the behaviour of a holomorphic self-map of the disk near a point of the boundary. In this talk we deal with the generalization of this concept to several complex variables: horospheres. The existence of horospheres in bounded strongly convex domains of $\mathbb{C}^n$ was proved by Abate in 1988 using Lempert’s theory of complex geodesics. It is difficult to generalize such proof to bounded strongly pseudoconvex domains, which is the natural class of domains to study in this context.

In this talk I will show how to obtain this generalization following a different route, that is, proving that the horofunction compactification of the domain is topologically equivalent to its Gromov compactification. This is a joint work with Matteo Fiacchi, Sébastien Gontard, and Lorenzo Guerini.