On Kiyoshi Oka's Unpublished Papers in 1943

TL;DR

This paper provides an English translation and analysis of Kiyoshi Oka's 1943 unpublished papers, which solved the longstanding Levi Problem for unramified Riemann domains over complex n-space, a major milestone in complex analysis.

Contribution

It presents the first English translation of Oka's last and most significant 1943 paper, clarifying his solution to the Levi Problem and its historical context.

Findings

Oka's affirmative solution to the Levi Problem for unramified Riemann domains over ^n

Historical insight into Oka's unpublished 1943 work

Discussion of an open problem left by Oka

Abstract

In 1943 from September to December Kiyoshi Oka wrote a series of papers numbered from VII to XI, as the research reports to Teiji Takagi (then, Professor of Tokyo Imperial University), in which he solved affirmatively the so-called Levi Problem (Hartogs' Inverse Problem termed by Oka) for unramified Riemann domains over $\C^n$. This problem which had been left open for more than thirty years then, was the last one of the Three Big Problems summarized by Behnke--Thullen 1934. The papers were hand-written in Japanese, consist of pp.~108 in total, and have not been published by themselves. The aim of the present article is to provide an English translation of the most important, last paper (Part II) with preparation (Part I). At the end of Part I we will discuss a problem which K. Oka left and is still open.