  The Geometric Shafarevich Conjecture and the Theorem of de Franchis state the
finiteness of the number of certain holomorphic objects on closed or punctured
Riemann surfaces. The analog of these kind of theorems for Riemann surfaces of
second kind is an estimate of the number of irreducible holomorphic objects up
to homotopy (or isotopy, respectively). This analog can be interpreted as a
quantitatve statement on the limitation for Gromov's Oka principle. For any
finite open Riemann surface $X$ (maybe, of second kind) we give an effective
upper bound for the number of irreducible holomorphic mappings up to homotopy
from $X$ to the twice punctured complex plane, and an effective upper bound for
the number of irreducible holomorphic torus bundles up to isotopy on such a
Riemann surface. The bound depends on a conformal invariant of the Riemann
surface. If $X_{\sigma}$ is the $\sigma$-neighbourhood of a skeleton of an open
Riemann surface with finitely generated fundamental group, then the number of
irreducible holomorphic mappings up to homotopy from $X_{\sigma}$ to the twice
punctured complex plane grows exponentially in $\frac{1}{\sigma}$.

Complex Variables