# A unified approach to the theory of separately holomorphic mappings

## TL;DR

This paper develops a unified theoretical framework for separately holomorphic mappings between complex analytic spaces, utilizing advanced techniques like Poletsky theory, Rosay Theorem, and conformal mappings to achieve global extensions from local data.

## Contribution

It introduces a novel unified approach to separately holomorphic mappings, combining multiple advanced methods and avoiding classical orthogonal basis techniques.

## Key findings

- Unified approach to separately holomorphic mappings
- Extension from local to global mappings demonstrated
- New techniques in conformal mappings and extremal functions

## Abstract

We extend the theory of separately holomorphic mappings between complex analytic spaces. Our method is based on Poletsky theory of discs, Rosay Theorem on holomorphic discs and our recent joint-work with Pflug on cross theorems in dimension 1. It also relies on our new technique of conformal mappings and a generalization of Siciak's relative extremal function. Our approach illustrates the unified character: ``From local informations to global extensions". Moreover, it avoids systematically the use of the classical method of doubly orthogonal bases of Bergman type.