Uniqueness Results for a Class of Holomorphic Mappings on a Complex Disc



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This dissertation gives a brief exposition of the history of Value Distribution Theory, often times referred to as Nevanlinna theory, and studies the case for Nevanlinna theory for holomorphic maps where the source is a disc developed by Ru-Sibony [16]. We start with a motivation into the subject and lay out some classical formulations with a focus on applications to the shared value problems. These problems are also referred to as uniqueness theorems. It is known that if two complex polynomials P and Q share two values without counting multiplicities, then they are the same. Such problem is called the shared value problem. In this dissertation, we focus on the study of the shared value problem for holomorphic maps where the source is a disc. There are derivations for several new unicity results for a class of holomorphic mappings from the disc into compact Riemann surfaces and n-dimensional complex projective space.



Nevanlinna theory, Complex geometry


Portions of this document appear in: Ru, Min, and Richard Walden. "Uniqueness Results for Holomorphic Mappings on the Disc." Acta Mathematica Vietnamica: 1-11.