Degeneracy theorems for meromorphic mappings of complete Kähler manifolds sharing hyperplanes in projective spaces

Let $M$ be a complete Kähler 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 $\mathbb{P}^n(\mathbb{C})$ $(n\ge 2)$ satisfy the condition $(C_\rho)$ and share $q$ $(q＞ C+\rho K)$ hyperplanes in general position regardless of multiplicity with certain positive constants $K$ and $C＜2n$ (explicitly estimated), then there are some algebraic relations between them. A degeneracy theorem for the product of $k$ $(2\le k\le n+1)$ meromorphic mappings sharing hyperplanes is also given. Our results generalize the previous results in the case of meromorphic mappings from $\mathbb{C}^m$ into $\mathbb{P}^n(\mathbb{C})$.