Banach Analytic Sets and a Non-Linear Version of the Levi Extension Theorem

TL;DR

This paper establishes a non-linear extension of the Levi theorem, showing that meromorphic functions extend along arbitrary complex curves, forming an infinite-dimensional analytic family, even without finite-dimensional constraints.

Contribution

It introduces a non-linear version of the Levi extension theorem for meromorphic functions along arbitrary complex curves, expanding the classical linear framework.

Findings

Meromorphic functions extend along arbitrary complex curves.

Extension occurs within an infinite-dimensional analytic family.

The domain of extension is a pinched domain filled by the family.

Abstract

We prove a certain non-linear version of the Levi extension theorem for meromorphic functions. This means that the meromorphic function in question is supposed to be extendable along a sequence of complex curves, which are arbitrary, not necessarily straight lines. Moreover, these curves are not supposed to belong to any finite dimensional analytic family. The conclusion of our theorem is that nevertheless the function in question meromorphically extends along an (infinite dimensional) analytic family of complex curves and its domain of existence is a pinched domain filled in by this analytic family.