Algebraic degeneracy and Uniqueness Theorems for Holomorphic Curves with Infinite Growth Index from a Disc into Pn(C) Sharing 2n+2 Hyperplanes

Let f and g be two holomorphic curves of a ball Delta(R) into P-n(C) with finite growth index, and let H-1,...,H2n+2 be 2n+2 hyperplanes in general position. In this paper, our first purpose is to show that if f and g have the same inverse images for all Hi(1 <= i <= 2n+2) with multiplicities counted to level li satisfying an explicitly estimate concerning c(f) and c(g), then the map fxg into P-n(C)xP(n)(C) must be algebraically degenerated. Our second purpose is to prove that f=g if they share 2n+2 hyperplanes with some certain conditions (in particular, they share 2n+2 hyperplanes with multiplicities counted to level n+1). Our results extend and improve the previous results for the case of holomorphic curve from C on these directions.