Degeneracy theorems for meromorphic mappings of a complete K\"{a}hler manifold sharing hyperplanes in a projective space

Let $M$ be a complete K"{a}hler manifold, whose universal covering is biholomorphic to a ball ${\mathbb{B}}^m(R_0)$ in ${\mathbb{C}}^m$ ($0<R_0\le +\infty$)\mathrm{.}Inthisarticle,wewillshowthatifthreemeromorphicmappings$f^1,f^2,f^3$of$M$into$\mathbb P^n(\mathbb C)\ (n\ge 2)$ satisfying the condition $(C_{\rho })$ and sharing $q(q>C+\rho K)$ hyperplanes in general position regardless of multiplicity with certain positive constants $K$ and $C <2n$ (explicitly estimated), then there are some algebraic relation between them. A degeneracy theorem for $k(2\le k\le n+1)$ meromorphic mappings sharing hyperplanes is also given. Our result generalize the previous result in the case where the mappings from ${\mathbb{C}}^m$ into ${\mathbb{P}}^n(\mathbb{C})$.