The Quasi-Additivity Law in conformal geometry

Abstract

On a Riemann surface S of finite type containing a family of N disjoint disks Di ("islands"), we consider several natural conformal invariants measuring the distance from the islands to ∂S and the separation between different islands. In a near degenerate situation we establish a relation between them called the Quasi-Additivity Law. We then generalize it to a Quasi-Invariance Law providing us with a transformation rule of the moduli in question under covering maps. This rule (and in particular, its special case called the Covering Lemma) has important applications in holomorphic dynamics.