Nevanlinna's five-value theorem on non-positively curved complete K\"ahler manifolds

TL;DR

This paper generalizes Nevanlinna's five-value theorem from complex plane functions to non-positively curved complete K"ahler manifolds, establishing conditions under which two meromorphic functions are identical based on shared values.

Contribution

It extends the classical five-value theorem to a broader geometric setting using algebraic dependence and growth conditions.

Findings

Two nonconstant meromorphic functions sharing five values are identical under certain conditions

The generalization applies to non-compact complete K"ahler manifolds with non-positive curvature

The approach involves algebraic dependence and growth condition analysis.

Abstract

Nevanlinna's five-value theorem is well-known as a famous theorem in value distribution theory, which asserts that two non-constant meromorphic functions on $\mathbb C$ are identical if they share five distinct values ignoring multiplicities in $\overline{\mathbb C}.$ The central goal of this paper is to generalize Nevanlinna's five-value theorem to non-compact complete K\"ahler manifolds with non-positive sectional curvature by means of the theory of algebraic dependence. With a certain growth condition imposed, we show that two nonconstant meromorphic functions on such class of manifolds are identical if they share five distinct values ignoring multiplicities in $\overline{\mathbb C}.$