The disk property. A short survey

We present some results obtained over the years regarding the disk property for complex manifolds and its connections with pseudoconvexity. The idea to use holomorphic disks to study domains of holomorphy in C goes back all the way to F. Hartogs [12] at the beginning of the twentieth century. Hartogs’ result was extended by Osgood [19] who proved what is called ”Hartogs extension theorem” stating that if Ω is a domain in C, K ⊂ Ω is a compact subset and Ω \K is connected then any holomorphic function on Ω \K can be extended to Ω. Osgood proof of Hartogs’ theorem is incomplete. For a complete proof along Hartogs’ ideas see [17]. The shortest and the most elegant proof is due to Ehrenpreis [9] and uses the ∂-method. Actually Hartogs extension theorem holds in a far more generally context, namely for cohomologically (n − 1)-complete normal complex spaces, see [4], [18], and [20]. A few years after Hartogs’ paper, E. E. Levi [16] founded the theory of pseudoconvexity and proved, using holomorphic disks, that a domain of holomorphy with smooth boundary must be pseudoconvex. Here we say that Ω is pseudoconvex if it has a strictly plurisubharmonic (i.e. its Levi form, or its complex Hessian, is positive definite) exhaustion function. Of course, by the Oka’s solution to the Levi problem, a domain in C is a domain of holomorphy if and only if it is pseudoconvex. One of the basic notions needed to deal with holomorphic disks for domains in C is the continuity principle. We denote by ∆ the open unit disk in C and, in general, by ∆r ⊂ C the disk of radius r centered at the origin.