Weakly complete complex surfaces

A weakly complete space is a complex space that admits a (smooth) plurisubharmonic exhaustion function. In this paper, we classify those weakly complete complex surfaces for which such an exhaustion function can be chosen to be real analytic: they can be modifications of Stein spaces or proper over a non-compact (possibly singular) complex curve, or foliated with real-analytic Levi flat hypersurfaces which in turn are foliated by dense complex leaves (these we call "surfaces of Grauert type"). In the last case, we also show that such Levi flat hypersurfaces are in fact level sets of a global proper pluriharmonic function, up to passing to a holomorphic double cover of the space. Our method of proof is based on the careful analysis of the level sets of the given exhaustion function and their intersections with the \emph{minimal singular set}, that is, the set where every plurisubharmonic exhaustion function has a degenerate Levi form