The precise form of Ahlfors' second fundamental theorem

TL;DR

This paper develops a method to determine the exact constant in Ahlfors' Second Fundamental Theorem for covering surfaces over the sphere, advancing the understanding of value distribution in complex analysis.

Contribution

It introduces a novel approach to identify the precise value of the constant in Ahlfors' SFT and provides a formula for its computation, building on historical estimates.

Findings

Existence of extremal surfaces is proven.

A formula for the exact value of the constant is derived.

The method improves precision in Ahlfors' SFT analysis.

Abstract

Ahlfors Second Fundamental Theorem of covering surfaces over the Riemann sphere are one of the major events in the history of function theory, which is a geometrical interpretation of the famous Nevanlinna's Second Fundamental Theorem in the theory of value distribution. The goal of this paper is to present a method to identify the precise value for that constant h in Ahlfors' SFT for covering surfaces over the unit sphere. This problem can be traced back to the early 1940s, when J. Dufresnoy first gave a numerical estimate of h for covering surfaces over a sphere. We will first prove the existence of extremal surfaces, and then we will give a formula to compute the precise value.