A Brief Chronicle of the Levi (Hartogs' Inverse) Problem, Coherence and an Open Problem

TL;DR

This paper reviews the historical development of the Levi (Hartogs' Inverse) Problem, coherence, and their solutions, highlighting overlooked contributions and discussing an open problem in the ramified case.

Contribution

It provides a chronological summary of key developments, including previously unpublished work by K. Oka, and discusses an open problem in the ramified case.

Findings

Oka's 1943 unpublished solution for unramified Riemann domains

Independent solutions by Bremermann and Norguet in 1954

Identification of an open problem in the ramified case

Abstract

Here we chronologically summarize briefly the developments of the Levi (Hartogs' Inverse) Problem together with the notion of coherence and its solution, shedding light on some records which have not been discussed in the past references. In particular, we will discuss K. Oka's unpublished papers 1943 which solved the Levi (Hartogs' Inverse) Problem for unramified Riemann domains of arbitrary dimension $n \geq 2$, usually referred as it was solved by Oka IX in 1953, H.J. Bremermann and F. Norguet in 1954 for univalent domains, independently. At the end we emphasize an open problem in a ramified case.