The precise form of Ahlfors' second fundamental theorem

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.