Improvements to the Montel-Carath\'eodory Theorem for families of $\mathbb{P}^n$-valued holomorphic curves

TL;DR

This paper improves the Montel-Carathéodory Theorem by establishing new sufficient conditions for the normality of families of holomorphic maps into complex projective space, enhancing understanding of their convergence properties.

Contribution

It provides novel sufficient conditions for normality of families of $ ext{P}^n$-valued holomorphic curves, extending classical results in complex analysis.

Findings

New sufficient conditions for normality of holomorphic families.

Enhanced criteria improve upon classical Montel-Carathéodory results.

Results applicable to families of holomorphic curves in projective space.

Abstract

In this paper, we establish various sufficient conditions for a family of holomorphic mappings on a domain $D\subseteq\mathbb{C}$ into $\mathbb{P}^n$ to be normal. Our results are improvements to the Montel-Carath\'eodory Theorem for a family of $\mathbb{P}^n$-valued holomorphic curves.