Algebraic dependence and finiteness problems of differentiably nondegenerate meromorphic mappings on K\"{a}hler manifolds

Let $M$ be a complete K\"{a}hler manifold, whose universal covering is biholomorphic to a ball $\mathbb B^m(R_0)$ in $\mathbb C^m$ ($0<R_0\le +\infty$). Our first aim in this paper is to study the algebraic dependence problem of differentiably meromorphic mappings. We will show that if $k$ differentibility nondegenerate meromorphic mappings $f^1,\ldots,f^k$ of $M$ into $\mathbb P^n(\mathbb C)\ (n\ge 2)$ satisfying the condition $(C_\rho)$ and sharing few hyperplanes in subgeneral position regardless of multiplicity then $f^1\wedge\cdots\wedge f^k\equiv 0$. For the second aim, we will show that there are at most two different differentiably nondegenerate meromorphic mappings of $M$ into $\mathbb P^n(\mathbb C)$ sharing $q\ (q\sim 2N-n+3+O(\rho))$ hyperplanes in $N-$subgeneral position regardless of multiplicity. Our results generalize previous finiteness and uniqueness theorems for differentiably meromorphic mappings of $\mathbb C^m$ and extend some previous results for the case of mappings on K\"{a}hler manifold.