Let $S$ be a riemann surface.
If S has idea boundary curves,then the intrinsic metric on $S$ can be defined by the restriction to $S$ of poincare metric of the double of $S$.
Also this metric can be derived from the restriction to $S$ of the poincare metric of $S^N$,where $S^N$ is the Nielsen extension of $S$.I don't know what the Nielsen extension of $S$ is



Nielsen extension of Riemann surface
The below is a cut and paste from a paper from Noemi Goldberg, PAMS, 1986. I blame Preview for the typesetting quality (= not knowing about mathjax). However, the original poster could also have googled "nielsen extension".


Let So be a Riemann surface of genus g with n punctures and m holes. Assume that 6<7- 6 + 2n + 3m > 0 and that m > 0. Then So can be represented as U/G where U is the upper half-plane and G is a torsion-free Fuchsian group of the second kind. There is an infinite set I of open intervals of R U {oo} on which G acts discontinuously. Let fi = U U L U I, where L denotes the lower half-plane. The quotient S$ = U/G is the double of So- It is a compact surface, except for finitely many punctures. The surface Sq can be represented as U/G', where G' is a torsion-free Fuchsian group of the first kind. Let D be a component of the preimage of So under the natural projection U —>U/G' and let Gi be the stabilizer of D in G'. Then the surface Si = U/Gi is the Nielsen extension of So- Given a surface So, let Sfebe the Nielsen extension of Sk-i, for k G N. Define the infinite Nielsen extension of So to be Sqo= SoUSi US2U•••.


