Meromorphic mappings on Kähler manifolds weakly sharing hyperplanes in ${\mathbb{P}}^n(\mathbb{C})$

In this paper, we study the uniqueness problem for linearly nondegenerate meromorphic mappings from a Kähler manifold into ${\mathbb{P}}^n(\mathbb{C})$ satisfying a condition $(C_{\rho })$ and sharing hyperplanes in general position, where the condition that two meromorphic mappings $f,g$ have the same inverse image for some hyperplanes $H$ is replaced by a weaker one that $f^{-1}(H)\subset g^{-1}(H)$. Moreover, we also give some improvements on the uniqueness problem and algebraic dependence problem of meromorphic mappings which share hyperplanes and satisfy $(C_{\rho })$ conditions for different non-negative numbers $\rho$.