On a hyperconvex manifold without non-constant bounded holomorphic functions

TL;DR

This paper presents an example of a hyperconvex manifold that lacks non-constant bounded holomorphic functions, constructed as a domain with a real-analytic Levi-flat boundary within a projective surface.

Contribution

It provides the first known example of such a hyperconvex manifold, expanding understanding of complex manifolds with specific boundary properties.

Findings

Existence of a hyperconvex manifold without non-constant bounded holomorphic functions

Construction as a domain with Levi-flat boundary in a projective surface

Implications for complex analysis and geometry of manifolds

Abstract

An example is given of a hyperconvex manifold without non-constant bounded holomorphic functions, which is realized as a domain with real-analytic Levi-flat boundary in a projective surface.