The Quasi-Additivity Law in conformal geometry

Abstract

On a Riemann surface $S$ of finite type containing a family of $N$ disjoint disks $D_i$ (“islands”), we consider several natural conformal invariants measuring the distance from the islands to $\partial S$ 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.

Authors

Jeremy Kahn

Department of Mathematics
Stony Brook University
Stony Brook, NY 11794
United States
and
Department of Mathematics
University of Toronto
Toronto, Ontario