Nevanlinna theory via holomorphic forms

TL;DR

This paper reformulates Nevanlinna theory for meromorphic functions using holomorphic forms, extending it to Riemann surfaces and generalizing classical results to new geometric contexts.

Contribution

It introduces a novel formulation of Nevanlinna functions via holomorphic forms and extends the theory to weak -exhausted Riemann surfaces.

Findings

Formulation of Nevanlinna functions using holomorphic forms.

Extension of Nevanlinna theory to Riemann surfaces.

Generalization of classical results to new geometric settings.

Abstract

This paper re-develops the Nevanlinna theory for meromorphic functions on $\mathbb C$ in the viewpoint of holomorphic forms. According to our observation, Nevanlinna's functions can be formulated by a holomorphic form. Applying this thought to Riemann surfaces, one then extends the definition of Nevanlinna's functions using a holomorphic form $\mathscr S$. With the new settings, an analogue of Nevanlinna theory on \emph{weak $\mathscr S$-exhausted Riemann surfaces} is obtained, which is viewed as a generalization of the classical Nevanlinna theory on $\mathbb C$ and $\mathbb D.$