Degeneracy and finiteness problems for holomorphic curves from a disc into Pⁿ(C) with finite growth index

Abstract

Let f¹, f², f³ are three holomorphic curves from a complex disc Δ(R) into Pⁿ(C) (n ≥ 2) with finite growth indexes c_f¹, c_f², c_f³ and sharing q (q ≥ 2n + 2) hyperplanes in general position regardless of multiplicity. In this paper, we will show the above bounds for the sum c_f¹ + c_f² + c_f³ to ensure that f¹ ∧ f² ∧ f³ = 0 or there are two curves among {f¹, f², f³} coincide to each other. Our results are generalizations of the previous degeneracy and finiteness results for linearly non-degenerate meromorphic mappings from C^m into Pⁿ(C) sharing (2n + 2) hyperplanes regardless of multiplicities.