Nevanlinna Theory on Complete Kähler Connected Sums With Non-parabolic Ends

Abstract

Motivated by invalidness of Liouville property for harmonic functions on the connected sum $\#^\vartheta\mathbb C^m$ with $\vartheta\geq2,$ we study Nevanlinna theory on a complete K\"ahler connected sum $$M=M_1\#\cdots\# M_\vartheta$$ with $\vartheta$ non-parabolic ends. Based on the global Green function method, we extend the second main theorem of meromorphic mappings to $M.$ As a consequence, we obtain a Picard's little theorem provided that all $M_j^,s$ have non-negative Ricci curvature, which states that every meromorphic function on $M$ reduces to a constant if it omits three distinct values.In particular, it implies that Cauchy-Riemann equation supports a rigidity of Liouville property as an invariant under connected sums.