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Normal Families of Holomorphic Curves and Sharing of Moving Hyperplanes Wandering on $\mathbb{P}^n$

Gopal Datt, Naveen Gupta, Nikhil Khanna, Ritesh Pal

Abstract

In this paper, we extend a result of Schwick concerning normality and sharing values in one complex variable for families of holomorphic curves taking values in $\mathbb{P}^n$. We consider wandering moving hyperplanes (i.e., depending on the respective holomorphic curve in the family under consideration), and establish a sufficient condition of normality concerning shared hyperplanes.

Normal Families of Holomorphic Curves and Sharing of Moving Hyperplanes Wandering on $\mathbb{P}^n$

Abstract

In this paper, we extend a result of Schwick concerning normality and sharing values in one complex variable for families of holomorphic curves taking values in . We consider wandering moving hyperplanes (i.e., depending on the respective holomorphic curve in the family under consideration), and establish a sufficient condition of normality concerning shared hyperplanes.

Paper Structure

This paper contains 4 sections, 3 theorems, 35 equations.

Table of Contents

  1. Introduction and main results
  2. Basic notions
  3. Essential Lemma
  4. Proof of Main Theorem

Key Result

Theorem 1.5

Let $\mathcal{F}$ be a family of functions holomoprhic in a domain $D\subset \mathbb{C}$ into $\mathbb{P}^n$. Let $\varepsilon$ be a positive real number in the open interval $(0, 1)$. Assume that for each $f\in \mathcal{F}$ there exists $2n+1$ moving hyperplanes $H_{1,f}, \dots, H_{(2n+1),f}$ such Suppose that for each $f\in \mathcal{F}$ Then $\mathcal{F}$ is normal in $D$.

Theorems & Definitions (7)

  • Definition 1.3
  • Theorem 1.5
  • Definition 2.1
  • Lemma 3.1: a special case of AladroKrantz
  • Remark 3.2
  • Lemma 3.3: Green
  • proof : Proof of Theorem \ref{['T: MainTheorem']}