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

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.