Let $X$ be a Riemann surface with analytic boundary. Assume that $X$ has negative Euler characteristic. Then there exists a conformal hyperbolic metric $X$ such that $\partial X$ consists of geodesics (conformal in the sense of being compatible with the given complex structure on $X$).


Does anyone has a reference (or even better, a quick proof) of this result?


Edit: Let me state a stronger version. Suppose $X$ is embedded in an open Riemann surface $Y$. Then there exists a conformal hyperbolic metric on a neighborhood of $X$ in $Y$ such that $\partial X$ consists of geodesics. Is this true?



Finding a hyperbolic metric with geodesic boundary on a given Riemann surface
A good reference is


W. Abikoff, The real analytic theory of Teichmuller space, Springer, 1980. (Chap. II section 1).


The idea is that you construct the double: it is the result of gluing of your surface with its mirror image. This is a compact surface, it has a hyperbolic metric, and the restriction of this metric on the original surface is the hyperbolic metric with geodesic boundary.


