Nevanlinna theory via holomorphic forms

Xianjing Dong and Shuangshuang Yang

Vol. 319 (2022), No. 1, 55–74

DOI: 10.2140/pjm.2022.319.55

Abstract

This paper redevelops Nevanlinna theory for meromorphic functions on $ℂ$ 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 for the $\mathcal{𝒮}$-exhausted Riemann surfaces is obtained, which is viewed as a generalization of the classical Nevanlinna theory for $ℂ$ and $\mathbb{𝔻}$.

Keywords

Nevanlinna theory, value distribution, second main theorem, defect relation, Riemann surface

Mathematical Subject Classification

Primary: 30D35

Milestones

Received: 4 January 2022

Revised: 26 April 2022

Accepted: 27 May 2022

Published: 28 August 2022