Elementary approach to the Hartogs extension theorem

TL;DR

This paper offers a self-contained, elementary proof of Hartogs' extension theorem in several complex variables, relying solely on basic complex analysis principles and avoiding advanced tools.

Contribution

It provides a new, accessible proof of Hartogs' theorem by establishing key theorems and simplifying the original approach.

Findings

Proof of higher-dimensional identity principle for holomorphic functions

Proof of Cauchy's integral formula for compact sets

Complete, self-contained exposition of Hartogs' extension theorem

Abstract

In this paper we present a proof of Hartogs' extension theorem, following T. Sobieszek's paper from 2003. Hartogs' theorem provides a large class of domains where holomorphic functions have analytic continuation to larger domains, and is "a several complex variables theorem" in nature because its conclusion is false in the complex plane. Sobieszek's proof is quite remarkable because he uses, stated in his paper without proofs, only higher-dimensional identity principle for holomorphic functions and Cauchy's integral formula for compact sets. We proved this two theorems here, making this exposition self-contained. The only background required is an undergraduate course in real and complex analysis and in point-set topology.