Meromorphic mappings of a complete connected K\"{a}hler manifold into a projective space sharing hyperplanes

Abstract

Let $M$ be a complete K\"{a}hler manifold, whose universal covering is biholomorphic to a ball $\mathbb B^m(R_0)$ in $\mathbb C^m$ ($0<R_0\le +\infty$). In this article, we will show that if three meromorphic mappings $f^1,f^2,f^3$ of $M$ into $\mathbb P^n(\mathbb C)\ (n\ge 2)$ satisfying the condition $(C_\rho)$ and sharing $q\ (q> 2n+1+\alpha+\rho K)$ hyperplanes in general position regardless of multiplicity with certain positive constants $K$ and $\alpha <1$ (explicitly estimated), then $f^1=f^2$ or $f^2=f^3$ or $f^3=f^1$. Moreover, if the above three mappings share the hyperplanes with mutiplicity counted to level $n+1$ then $f^1=f^2=f^3.$ Our results generalize the finiteness and uniqueness theorems for meromorphic mappings of $\mathbb C^m$ into $\mathbb P^n(\mathbb C)$ sharing $2n+2$ hyperplanes in general position with truncated multiplicity.