Movement of branch points in Ahlfors' theory of covering surfaces

TL;DR

This paper proves a key result related to the movement of branch points in Ahlfors' theory, which supports the existence of extremal surfaces and sharpens the Second Fundamental Theorem in covering surface theory.

Contribution

It introduces a new result on branch point movement that underpins the existence of extremal surfaces and refines Ahlfors' Second Fundamental Theorem.

Findings

Proves a movement result for branch points in Ahlfors' theory

Supports the existence of extremal surfaces in covering surface problems

Provides a sharper form of Ahlfors' Second Fundamental Theorem

Abstract

In this paper, we will prove a result which is used by Guang-Yuan Zhang in another paper in which the existence of extremal surfaces for covering surfaces is proved and the sharp form of Ahlfors' Second Fundamental Theorem is given.