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