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?

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?