Normality criterion for a family of holomorphic curves that partially share wandering hyperplanes with their derivatives, and holomorphic functions lifted to curves in $P^2(\mathbb{C})$

TL;DR

This paper extends normality criteria for families of holomorphic curves and functions in complex projective spaces, introducing new conditions involving shared wandering hyperplanes and derivatives, with implications for Bloch's principle.

Contribution

It generalizes previous results on normality by incorporating partial sharing of wandering hyperplanes and derivatives, and provides a new representation of holomorphic functions in projective space.

Findings

Established a new normality criterion for holomorphic curves in $P^N( ext{C})$.

Derived a normality criterion for meromorphic functions sharing wandering holomorphic functions.

Constructed a counterexample to the converse of Bloch's principle.

Abstract

In this paper we generalize a result of Ye, Pang and Yang[12] on the normality of a family of holomorphic curves in $P^N(\mathbb{C})$. Further we obtain a normality criterion for family of meromorphic functions that partially share wandering holomorphic functions with their derivatives. We also devise a tractable representation of complex valued holomorphic functions from D as functions from D to $P^2(\mathbb{C})$ obtain a normality criterion that leads to a counterexample to the converse of Bloch's principle.