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Meromorphically normal families and a meromorphic Montel-Carathéodory theorem

Gopal Datt

Abstract

In this paper, we present various sufficient conditions for a family of meromorphic mappings on a domain $D\subset \mathbb{C}^m$ into $\mathbb{P}^n$ to be meromorphically normal. Meromorphic normality is a notion of sequential compactness in the meromorphic category introduced by Fujimoto. We give a general condition for meromorphic normality that is influenced by Fujimoto's work. The approach to proving this result allows us to establish meromorphic analogues of several recent results on normal families of $\mathbb{P}^n$-valued holomorphic mappings. We also establish a meromorphic version of the Montel-Carathéodory theorem.

Meromorphically normal families and a meromorphic Montel-Carathéodory theorem

Abstract

In this paper, we present various sufficient conditions for a family of meromorphic mappings on a domain into to be meromorphically normal. Meromorphic normality is a notion of sequential compactness in the meromorphic category introduced by Fujimoto. We give a general condition for meromorphic normality that is influenced by Fujimoto's work. The approach to proving this result allows us to establish meromorphic analogues of several recent results on normal families of -valued holomorphic mappings. We also establish a meromorphic version of the Montel-Carathéodory theorem.

Paper Structure

This paper contains 8 sections, 8 theorems, 42 equations.

Table of Contents

  1. Introduction and Main Results
  2. Some notations
  3. Basic notions
  4. Some Examples
  5. Essential lemmas
  6. The proof of Theorem \ref{['T: mainthm2n+1']}
  7. The proofs of Theorem \ref{['T: finitefamily']} and its corollaries
  8. The proof of Theorem \ref{['T: mainthm3n+1']}

Key Result

Theorem 1.3

Let $D$ be a domain in $\mathbb{C}^m$. Let $\mathcal{F}$ be a family of meromorphic mappings from $D$ into $\mathbb{P}^n$, $\mathcal{G}$ a family of holomorphic mappings from $D$ into $\mathbb{P}^n$, and $\mathcal{H}$ a collection of hyperplanes in $\mathbb{P}^n$. Suppose that, for each $f\in \mathc Furthermore, assume that for any mapping in the closure of $\mathcal{G}$, its range is not a subset

Theorems & Definitions (20)

  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Corollary 1.6: Meromorphic Montel--Carathéodory Theorem
  • Theorem 1.7
  • Definition 2.3
  • Definition 2.4: Fujimoto74
  • Definition 2.5: Fujimoto74
  • Example 3.1
  • Example 3.2
  • ...and 10 more