Algebraic degeneracy and Uniqueness Theorems for Holomorphic Curves with Infinite Growth Index from a Disc into $${\mathbb {P}}^n({\mathbb {C}})$$ Sharing $$2n+2$$ Hyperplanes

Let f and g be two holomorphic curves of a ball $$\Delta (R)$$ into $${\mathbb {P}}^n({\mathbb {C}})$$ with finite growth index, and let $$H_1,\ldots ,H_{2n+2}$$ 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 $$H_i\ (1\le i\le 2n+2)$$ with multiplicities counted to level $$l_i$$ satisfying an explicitly estimate concerning $$c_f$$ and $$c_g$$ , then the map $$f\times g$$ into $${\mathbb {P}}^n({\mathbb {C}})\times {\mathbb {P}}^n({\mathbb {C}})$$ 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 $${\mathbb {C}}$$ on these directions.