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In previous works, G. Tomassini and the authors studied and classifiedcomplex surfaces admitting a real-analytic pluri-subharmonic exhaustionfunction; let $X$ be such a surface and $D\subseteq X$ a domain admitting a\emph{continuous} plurisubharmonic exhaustion function: what can be said aboutthe geometry of $D$? If the exhaustion of $D$ is assumed to be smooth, thesecond author already answered this question; however, the continuous case ismore difficult and requires different methods. In the present paper, we addresssuch question by studying the local maximum sets contained in $D$ and theirinterplay 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 geometricfeatures of $X$.

Domains with a continuous exhaustion in weakly complete surfaces