The precise form of Ahlfors' second fundamental theorem

GUang-Yuan Zhang

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arXiv · 2307.04623

The precise form of Ahlfors' second fundamental theorem

GUang-Yuan Zhang

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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.

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GUang-Yuan Zhang. 2023-07-10. The precise form of Ahlfors' second fundamental theorem. https://arxiv.org/abs/2307.04623

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