Normal families and quasiregular mappings

TL;DR

This paper extends the concept of normal families from holomorphic functions to quasiregular mappings in higher dimensions, providing a unified framework and new characterizations for various classes of quasiregular mappings.

Contribution

It generalizes Beardon and Minda's Lipschitz condition to higher dimensions and introduces a unified approach to analyze normality of quasiregular mappings using modulus of continuity.

Findings

Unified framework for normal quasiregular mappings

Characterizations of Yosida and Bloch quasiregular classes

Upper bounds on growth rates of these classes

Abstract

Beardon and Minda gave a characterization of normal families of holomorphic and meromorphic functions in terms of a locally uniform Lipschitz condition. Here, we generalize this viewpoint to families of mappings in higher dimensions that are locally uniformly continuous with respect to a given modulus of continuity. Our main application is to the normality of families of quasiregular mappings through a locally uniform H\"older condition. This provides a unified framework in which to consider families of quasiregular mappings, both recovering known results of Miniowitz, Vuorinen and others, and yielding new results. In particular, normal quasimeromorphic mappings, Yosida quasiregular mappings and Bloch quasiregular mappings can be viewed as classes of quasiregular mappings which arise through consideration of various metric spaces for the domain and range. We give several characterizations of these classes and obtain upper bounds on the rate of growth in each class.