Question

I have following two questions

1) Let $S$ be a compact oriented surface with (non-empty) boundary. Also assume that the Euler characteristic of $S$ is negative (Thus, $S$ is not disk or annulus). Then is it possible to put a hyperbolic metric on $S$ such that every boundary component becomes a geodesic?

2) In above case, what is the universal cover? Is it a subset of the unit disk with hyperbolic metric?

Answer 1

The answers to both questions are positive.

1) If the Euler characteristic of $S$ is negative, then you can find a pants decomposition (where some of the pants boundary components are glued together, while the others are boundary components of $S$). Then you can put hyperbolic metrics on each pair of pants, while prescribing the lengths of the boundary component, so that you can glue them into a hyperbolic metric of $S$ (and you have parameters left for the gluing, just like for a closed surface).

2) You can always, by gluing additional pants, consider $S$ endowed with a hyperbolic metric constructed as above, as a subsurface with geodesic boundary of a closed hyperbolic surface $\bar S$. Then the universal cover of $\bar S$ is the Poincaré disc, and the boundary components of $S$ lift to complete geodesics. The universal cover of $S$ is then one of the connected components of the complement of these complete geodesics.