Branch values in Ahlfors' theory of covering surfaces

TL;DR

This paper proves the existence of a modified covering surface with branch values confined to a specified set, simplifying the analysis in Ahlfors' theory of covering surfaces and extending previous results.

Contribution

It generalizes key lemmas by constructing a new surface with controlled branch values, aiding in the study of Ahlfors' second fundamental theorem.

Findings

Existence of a surface with branch values contained in E_q

The new surface satisfies area and length inequalities

Generalization of previous lemmas in Ahlfors' theory

Abstract

In the study of the constant in Ahlfors' second fundamental theorem involving a set E_{q} of q points, branch values of covering surfaces outside E_{q} bring a lot of troubles. To avoid this situation, for a given surface S, it is useful to construct a new surface So such that L(So) <=L(S), and H(S)>=H(S), and all branch values of So are contained in E_{q}. The goal of this paper is to prove the existence of such So, which generalizes Lemma 9.1 and Theorem 10.1 in Zhang G.Y.: The precise bound for the area-length ratio in Ahifors' theory of covering surfaces. Invent math 191:197-253(2013)