Summary: 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 \((C_p)\) 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 \(\mathbb{P}^n(\mathbb{C})\).