On weakly complete surfaces [Sur les surfaces faiblement complètes]

A weakly complete space is a (connected) complex space endowed with a (smooth) plurisubharmonic exhaustion function. In this paper, we classify the weakly complete surfaces (i.e. weakly complete manifolds of dimension 2) for which such exhaustion function can be chosen to be real analytic: they can be modifications of Stein spaces or proper (i.e. endowed with a proper surjective holomorphic map onto) a non-compact (possibly singular) complex curve or surfaces of Grauert type i.e. foliated with real analytic Levi flat hypersurfaces whose Levi foliation has dense complex leaves. 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 covering.