Value distribution of meromorphic mappings on complete K\"ahler connected sums with non-parabolic ends

TL;DR

This paper investigates the value distribution of meromorphic mappings on complete Kähler connected sums with non-parabolic ends, establishing a second main theorem and conditions for Liouville's property rigidity for harmonic functions.

Contribution

It introduces a second main theorem in Nevanlinna theory for these connected sums and identifies geometric and volume growth conditions ensuring Liouville's property.

Findings

Established a second main theorem for meromorphic mappings on the specified manifolds.

Proved Liouville's property rigidity under certain geometric and volume growth conditions.

Connected sums with non-parabolic ends influence harmonic function properties and value distribution.

Abstract

All harmonic functions on $\mathbb C^m$ possess Liouville's property, which is well-known as the Liouville's theorem. In 1979, Kuz'menko and Molchanov discovered a phenomenon that the Liouville's property is not rigid for some harmonic functions on the connected sum $\mathbb C^m\#\mathbb C^m,$ where there exist a large number of non-constant bounded harmonic functions. This discovery motivates us to explore conditions under which harmonic functions possess Liouville's property. In this paper, we discuss the value distribution of meromorphic mappings from complete K\"ahler connected sums with non-parabolic ends into complex projective manifolds. Under a geometric condition, we establish a second main theorem in Nevanlinna theory. As a consequence, we prove that the Cauchy-Riemann equation ensures the rigidity of Liouville's property for harmonic functions if such connected sums satisfy a volume growth condition.