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I am trying to figure out the details on giving a surface S a hyperbolic metric with geodesic boundary, i.e., a metric of constant sectional curvature -1 so that the (manifold) boundary components, i.e., a collection of disjoint simple-closed curves are geodesics under this metric. So far I know:

1) There are genus constraints for the surface. Does this have to see with Gauss-Bonnet?

2) Something; not sure exactly what, can be done by gluing pairs-of-pants http://en.wikipedia.org/wiki/Pair_of_pants_(mathematics [Broken])

but not fully sure how this works.

Not much more. Any ideas, refs., please?

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# Hyperbolic Manifold With Geodesic Boundary?

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