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. In our results, the condition that two meromorphic mappings $f,g$ have the same inverse image for some hyperplanes $H$ is replaced by a weaker one that $f^{-1}(H)\subseteq g^{-1}(H)$. Moreover, we also give some improvements on the uniqueness problem and the algebraic dependence problem of meromorphic mappings which share hyperplanes and satisfy $(C_\rho)$ conditions for different non-negative numbers $\rho$.