Distances invariantes et points fixes d'applications holomorphes

TL;DR

This paper establishes a contraction property of the Carathéodory metric on hyperbolic complex manifolds and investigates fixed points of holomorphic maps between such spaces.

Contribution

It proves a new contraction inequality for the Carathéodory metric on hyperbolic manifolds and explores fixed point results for holomorphic applications.

Findings

Existence of a contraction factor k<1 for the Carathéodory metric

Contraction inequality relating metrics on X and U

Results on fixed points of holomorphic mappings

Abstract

In this paper, we prove the following result : let X be a complex manifold, hyperbolic for the Carath\'eodory distance and let U be an open set relatively compact in X. Then, there exists k<1 such that we get, for the Carath\'eodory infinitesimal metric E_X(x,v) less or equal to kE_U(x,v). We also get results concerning fixed points of holomorphic mappings from X to U.