Recall a domain in the Riemann sphere is called a circle domain if every connected component of its boundary is either a circle or a point. Circle domain is interesting as Koebe asks whether any planar domain is conformal to a circle domain.
Now we focus on the circle domain of hyperbolic type (as a Riemann surface which is a quotient (by a discrete and proper action) of unit disk), then the standard hyperbolic metric on the unit disk descends onto the circle domain.
Question 1: Note that the induced hyperbolic metric is always complete, are there further estimates on this metric?
For the standard triply-punctured sphere, the asymptotic near each puncture is of a cusp. I found a nice picture of its geodesic, see here.
For a simply connected domain $\Omega \subset \mathbb{C}$, the hyperbolic metric $\rho(z)|dz|$ is equivalent to $\frac{1}{d(z, \partial \Omega)}|dz|$, as a corollary of Koebe $\frac{1}{4}$ theorem. See here.
Question 2: Function theory on some circle domain could be complicated. For example, on p.252 of Ahlfors-Sario the authors use some generalized Cantor set to show that its complement in the complex plane, as a Riemann surface, enjoy the property: any bounded holomorphic function is constant. I would appreciate if anyone suggests some related results, even some specific examples seem interesting.