Fenchel-Nielsen coordinates give a coordinatization of Teichmuller space for compact conformal surfaces admitting a pants decomposition. But not all compact conformal surfaces (possibly with boundary, which we assume totally geodesic) of genus less than 2 admit such a decomposition, such as the cylinder. Is there an extension of Fenchel-Nielsen coordinates to these cases?

As Lee points out, the only nontrivial examples are the cylinder and the torus. These admit Euclidean metric, which can be normalized to have area one, and for a cylinder we can assume that the boundary components are the same length and parallel. (in other words, metrically the cylinder has the form $a S^1 \times [0, 1/a]$). Then the Fenchel-Nielsen coordinate (there is only one) is $a,$ the length of the boundary curve. A torus can be made from a cylinder as above by gluing the two boundary components with a twist, so now we have both the length and the twist coordinate. It is pretty clear that this last construction is what gave rise to the hyperbolic version of Fenchel-Nielsen coordinates (I am not sure if this is actually written down somewhere).

punctureddisk compact? $\endgroup$ – Alex Degtyarev Feb 28 '14 at 18:47