When generalizing the Dirichlet Boundary Problem from the disk to a general Riemann Surface $X$, Forster solves to problem for:
on all domains Domains $D \in X$ which are relatively compact and that are contained in a chart $(U,z)$, so that $z(D) \subset \mathbb{C}$ is a disk.
The reasoning, according to Forster, is that harmonic functions are invariant under biholomorphic mappings.
So, I assume the way Forster is generalizing the problem, is by projecting it by a biholomorphic mapping onto a disk in $\mathbb{C}$, solving it there the way it was solved in the pages before, and then returning to the Riemann-Surface by using another biholomorphic mapping. Is that correct? There is never a meantioning on how the problem is generalized.
If it works the way I just described, then it makes sense to me, that we want $z(D)$ to be a disk. What I do not understand though, is why it is necessary for $D$ to be relatively compact.
What problems would arise if we did no assumed that $D$ is relatively compact?
An answer would really help me to understand the rest of the chapter.
The collection of harmonic and/or holomorphic functions on unbounded regions in $\mathbb C$ behaves significantly differently from those on a bounded region in $\mathbb C$.
The prototype for where we'd like to construct harmonic (or holomorphic) functions is the unit disk... as opposed to a disk with a part going out to infinity. Perhaps the Phragmen-Lindelof illustrates some aspect of this: the maximum modulus principle is not quite correct on a strip or half-strip... without a growth condition.