Quasiconformal Normalization of Random Meromorphic Functions

TL;DR

This paper introduces a method for normalizing random meromorphic functions using quasiconformal maps, demonstrating that the associated surfaces are almost surely parabolic and providing growth bounds for their Nevanlinna characteristic.

Contribution

It develops a normalization technique for random meromorphic functions via quasiconformal maps, establishing almost sure parabolicity and growth bounds.

Findings

Surfaces are almost surely parabolic.

Constructed quasiconformal maps are surjective and approximately linear.

Bounds on the growth order of Nevanlinna characteristic.

Abstract

We study the conformal type of surfaces spread over the sphere via random quasiconformal maps. Constructing a random Beltrami coefficient on the complex plane, we obtain a locally quasiconformal homeomorphism with prescribed dilatation that is almost surely surjective and, with high probability, approximately linear. This yields a normalization for random meromorphic functions associated to surfaces spread over the sphere, from which we prove that the surfaces are almost surely parabolic and obtain bounds on the growth order of their Nevanlinna characteristic.