The Second Main Theorem via Negative Curvature

Abstract

Shortly after R. Nevanlinna’s first proof of the Second Main Theorem, Nevanlinna’s brother, F. Nevanlinna, gave a “geometric” proof of the Second Main Theorem. In this chapter, we give a geometric proof of the Second Main Theorem based on “negative curvature,” which broadly speaking, has the same overall structure as F. Nevanlinna’s proof, although in terms of details, the proof we present here bears much greater resemblance to the work of Ahlfors [Ahlf 1941]. In Chapter 4, we will give another proof of the Second Main Theorem that is closer to R. Nevanlinna’s original proof. Of course, neither the Nevanlinna brothers nor Ahlfors were interested in the exact structure of the error term. The error term we will present here is essentially due to P.-M. Wong [Wong 1989].