The Second Main Theorem via Negative Curvature

  • William Cherry
  • Zhuan Ye
Part of the Springer Monographs in Mathematics book series (SMM)


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].


Error Term Meromorphic Function Negative Curvature Isoperimetric Inequality Subharmonic Function 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • William Cherry
    • 1
  • Zhuan Ye
    • 2
  1. 1.Department of MathematicsUniversity of North TexasDentonUSA
  2. 2.Department of Mathematical SciencesNorthern Illinois UniversityDeKalbUSA

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