Testing holomorphy on curves

TL;DR

The paper constructs a foliation of a domain in complex space into curves, demonstrating that if a function extends holomorphically near each curve, then it is holomorphic on the entire domain.

Contribution

It introduces a novel foliation approach in several complex variables to characterize holomorphic functions via curve extensions.

Findings

Constructed a continuous foliation of the domain into curves.

Proved that holomorphic extendability near each curve implies global holomorphicity.

Provides a new geometric criterion for holomorphy in complex domains.

Abstract

For a domain $D\subset {\Bbb{C}}^n$ we construct a continuous foliation of $D$ into one real dimensional curves such that any function $f\in {C^1(D)}$ which can be extended holomorphically into some neighborhood of each curve in the foliation will be holomorphic on $D$.