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#### Riemann surfaces of second kind and effective finiteness theorems

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https://doi.org/10.1007/s00209-022-03018-3

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Jöricke_Riemann surfaces of second kind and effective finiteness theorems_2022.pdf

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##### Citation

Jöricke, B. (2022). Riemann surfaces of second kind and effective finiteness theorems.* Mathematische Zeitschrift,* *302*(1), 73-127. doi:10.1007/s00209-022-03018-3.

Cite as: https://hdl.handle.net/21.11116/0000-000A-A96F-C

##### Abstract

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 $\frac{1}{\sigma}$.

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 $\frac{1}{\sigma}$.