# Domains with a continuous exhaustion in weakly complete surfaces

## TL;DR

This paper investigates the geometric structure of domains with continuous plurisubharmonic exhaustion functions in weakly complete surfaces, revealing that such domains share properties with the ambient surface unless they are modifications of Stein spaces.

## Contribution

It extends previous classifications to the continuous exhaustion case, analyzing local maximum sets and their relation to the complex geometry of the surface.

## Key findings

- Domains with continuous exhaustion functions often mirror the geometry of the ambient surface.
- If not a modification of a Stein space, the domain shares the same geometric features as the surface.
- The study introduces new methods for analyzing local maximum sets in this context.

## Abstract

In previous works, G. Tomassini and the authors studied and classified complex surfaces admitting a real-analytic pluri-subharmonic exhaustion function; let $X$ be such a surface and $D\subseteq X$ a domain admitting a \emph{continuous} plurisubharmonic exhaustion function: what can be said about the geometry of $D$? If the exhaustion of $D$ is assumed to be smooth, the second author already answered this question; however, the continuous case is more difficult and requires different methods. In the present paper, we address such question by studying the local maximum sets contained in $D$ and their interplay with the complex geometric structure of $X$; we conclude that, if $D$ is not a modification of a Stein space, then it shares the same geometric features of $X$.