Is the Poincaré metric continuous with respect to the domain?

Question

Suppose $K \subset \mathbb{C}$ is a Cantor set and let $u:\mathbb{C} \setminus K \to \mathbb{R}$ be the maximal smooth function such that the conformal metric $e^{2u}(\mathrm{d}x^2 + \mathrm{d}y^2)$ has constant curvature $-1$ on $\mathbb{C} \setminus K$.

Suppose we put the Hausdorff distance on the set of compact subsets of $\mathbb{C}$. I'm interested in how $u$ varies with respect to the Cantor set $K$.

Question: Does $u$ vary continuously in the smooth topology on compact subsets of $\mathbb{C} \setminus K$ with respect to $K$?

Some more background: The existence of a unique maximal hyperbolic metric (sometimes called the Poincaré metric of $\mathbb{C} \setminus K$) is proved for example in Ahlfors' book. I've looked around and found several references estimating the metric in terms of distance to the Cantor set (for example this one). I haven't found any result on how the metric varies with respect to the domain save the easy observation of domain monotonicity (i.e. $u$ grows if $K$ does).

Answer 1

In "D.A. Hejhal, Universal covering maps for variable regions, Mathematische Zeitschrift 137 (1974), 7--20." it is shown that the universal covering map of a hyperbolic domain depends locally uniformly continuously on the domain. The convergence of the derivatives follows from Cauchy's integral formula.

Since the hyperbolic metric on a domain is the push-forward of the Poincare metric by the universal covering map, the hyperbolic metric also depends continuously on the domain.

This is all with respect to the Caratheodory topology on pointed domains, which is a weaker topology than the Hausdorff topology on the complements, so it implies what you want.