Circle packing and Riemann surfaces (Q1803639)

A theorem of W. Thurston says that any given triangulation of a closed surface of genus \(g\) can be represented as the nerve of a certain circle packing, and that this circle packing determines a Riemann surface structure on \(S\) which is hyperbolic for \(g\geq 2\). The present paper generalizes this result to the case of compact bordered surfaces of genus \(g\geq 2\). The case \(g\leq 1\) was treated earlier. Furthermore the relationship is studied between polyhedral surfaces and hyperbolic metrics with isolated singularities and the index of such a singularity. For the existence problem with prescribed restriction at the border and prescribed indices at some interior points see a recent paper by \textit{B. Garrett} in Discrete Comp. Geom. (title: Circle packings and polyhedral surfaces).