Resumen de Horosphere topology

Filippo Bracci, Hervé Gaussier

• We introduce a prime end-type theory on complete Kobayashi hyperbolic manifolds using horosphere sequences. This allows to introduce a new notion of boundary – new even in the unit disc in the complex space – the horosphere boundary, and a topology on the manifold together with its horosphere boundary, the horosphere topology. We prove that a bounded strongly pseudoconvex domain endowed with the horosphere topology is homeomorphic to its Euclidean closure, while for the polydisc such a horosphere topology is not even Hausdorff and is different from the Gromov topology. We use this theory to study boundary behavior of univalent maps from bounded strongly pseudoconvex domains.