Algebraic dependence and finiteness problems of differentiably nondegenerate meromorphic mappings on Kähler manifolds

Let M be a complete Kähler manifold, whose universal covering is biholomorphic to a ball 𝔹^m(R₀) in ℂ^m (0 < R₀ +∞). Our first aim in this paper is to study the algebraic dependence problem of differentiably meromorphic mappings. We will show that if k differentibility nonde-generate meromorphic mappings f¹, …, f^k of M into ℙⁿ(ℂ) (n ≥ 2) satisfying the condition (C_ρ) and sharing few hyperplanes in subgeneral position regardless of multiplicity then f¹ Λ … Λ f^k0. For the second aim, we will show that there are at most two different differentiably nondegenerate meromorphic mappings of M into ℙⁿ(ℂ) sharing q (q ∼ 2N − n + 3 + O(ρ)) hyperplanes in N−subgeneral position regardless of multiplicity. Our results generalize previous finiteness and uniqueness theorems for differentiably meromorphic mappings of ℂ^m into ℙⁿ(ℂ) and extend some previous results for the case of mappings on Kähler manifold.