On Weakly 1-Complete Surfaces without Non-Constant Holomorphic Functions

In this paper, we are interested in a class of two dimensional complex manifolds which are called weakly 1-complete surfaces. Here we call a two dimensional complex manifold X a weakly 1-complete surface if X possesses a C°°-exhausting plurisubharmonic function. This class includes two different extreme objects: compact analytic surfaces and two dimensional Stein manifolds. But at the same time, this class includes some curious examples from the function theoretic point of view i.e. there are weakly 1-complete surfaces without non-constant holomorphic functions (see [3] [6] [8] [12]) and moreover non-compact weakly 1-complete surfaces have an extreme function theoretic property i.e. a non-compact weakly 1-complete surface X is holomorphically convex if and only if X possesses a non-constant holomorphic function (see [9]). Looking back to the case of compact analytic surfaces, roughly speaking, they are classified by the existence or non-existence of meromorphic function. Hence it is natural to suppose that this aspect might give a new standpoint to analyze such a curious example in the class of weakly 1-complete surfaces as far as weakly 1-completeness is expected as a nice intermediate concept between compactness and Stein. This note is an attempt towards the problem of the existence of meromorphic function on non-compact weakly 1-complete surfaces. From now on, all weakly 1-complete surfaces are connected and non-compact and have no exceptional compact curves of the first kind unless otherwise is explicitly stated. Then we shall prove the following theorem.