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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?

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  • $\begingroup$ The Teichmuller space for the punctured disc is a single point. $\endgroup$ – Adam Epstein Feb 28 '14 at 18:43
  • $\begingroup$ In what sense is punctured disk compact? $\endgroup$ – Alex Degtyarev Feb 28 '14 at 18:47
  • $\begingroup$ Thanks for the clarification - I meant the disc with a disc removed, i.e. the cylinder. $\endgroup$ – Jamie Vicary Feb 28 '14 at 18:52
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    $\begingroup$ The only compact surfaces without a pants decomposition are the sphere with zero, one, or two holes, and the torus. There sphere with zero or one hole has a unique conformal structure. The sphere with two holes, aka the cylinder, has conformal structures classified by the extremal length of the core curve: each conformal structure on the cylinder is conformally equivalent to a unique product $S^1 \times [0,d]$. Conformal structures on the torus are classified by the upper half plane. $\endgroup$ – Lee Mosher Feb 28 '14 at 22:10
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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).

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