I already asked this question on math.stackexchange, but it was suggested that I post it here as well.

The paper Devadoss, Heath, and Vipismakul - Deformations of bordered Riemann surfaces and associahedral polytopes defines bordered Riemann surfaces with geodesic boundaries, punctures, and marked points on the boundaries.

Then it is stated that

Indeed, any stable marked bordered Riemann surface has a unique hyperbolic metric such that it is compatible with the complex structure, where all the boundary circles are geodesics, all punctures are cusps, and all boundary marked points are half cusps.

I understand this statement for the boundaries and the punctures, but not for the marked points on the boundaries.

Can someone explain this to me, maybe with some neat example? Any reference with good examples?


1 Answer 1


This means that the boundary is geodesic with cusps in the marked points.

The easiest example is a disk with 3 marked points on its boundary. In this case the hyperbolic metric is given by identification with an ideal triangle in ${\mathbb H}^2$, i.e., a triangle with its three vertices in $\partial_\infty{\mathbb H}^2$ and the geodesics between these ideal vertices as its boundary.

  • $\begingroup$ And so an half cusp would be something like the infinite region of the poincare' disk bounded by two geodesics meeting at infinity? $\endgroup$ Feb 27, 2018 at 10:04
  • $\begingroup$ Could you make an example with two boundary components? I was thinking of removing a circle from the convex hull of some points in the boundary, but a circle is not a geodetic... $\endgroup$ Feb 27, 2018 at 10:05
  • 1
    $\begingroup$ Yes, you should think of the half cusp as a region bounded by two geodesics meeting at infinity. $\endgroup$
    – ThiKu
    Feb 27, 2018 at 10:15
  • $\begingroup$ Gluing several ideal triangles along some (but not all) of their edges you can produce examples of higher genus and with more boundary components. $\endgroup$
    – ThiKu
    Feb 27, 2018 at 10:16
  • $\begingroup$ This is related to the "shear coordinates" on the Teichmuller space, right? $\endgroup$ Feb 27, 2018 at 10:26

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