Normal Families: a Geometric Perspective

Abstract

In this largely expository paper we present an alternative to the common practice of discussing normal families of analytic maps in terms of the Euclidean metric and equicontinuity. Indeed, in most cases the hyperbolic metric and the Schwarz–Pick Lemma are available, and then equicontinuity is redundant and is replaced by a much stronger Lipschitz condition that is expressed in terms of conformally invariant metrics. Here, we discuss normal families in terms of (not necessarily analytic) maps that satisfy types of uniform Lipschitz conditions with respect to various conformal metrics, especially the hyperbolic and spherical metrics. A number of classical results for normal families of analytic maps extend to these broader classes of (not necessarily analytic) functions that satisfy types of uniform Lipschitz conditions.