Nevanlinna theory via holomorphic forms

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 $\mathcal{S}$. With the new settings, an analogue of Nevanlinna theory on \emph{weak $\mathcal{S}$-exhausted Riemann surfaces} is obtained, which is viewed as a generalization of the classical Nevanlinna theory on $\mathbb{C}$ and $\mathbb{D}.$

This is a final version, prepared for publication in Pacific Journal of Mathematics