Let $\mathbb{D}$ be the unit disc, $Y$ be a Riemann surface and $\pi:\mathbb{D}\longrightarrow Y$ be a universal covering map. Suppose that $\pi$ is not given explicitly. Under what conditions can it be deduced that $\pi$ is a local biholomorphism?
Alternatively, let $U\subset \mathbb{D}$ be a single sheet of $\pi$. Under what conditions is $\pi\mid_U$ holomorphic?
Obviously, $\pi$ is proper and a local homeomorphism. I'm wondering if there are any results (or else, any insightful observations) that may establish conditions under which $\pi$ must be locally holomorphic as well.
For context, the question arose while constructing a proof of Picard's Big Theorem using some results in Riemann Surface theory. Toward the end, I realized that being able to establish analyticity of an arbitrary covering map would be a powerful tool, but I couldn't find any results in Forster or Jost (maybe because it's unfeasible, but I'd like to think that there is at least a weak result).
Here is a solution in great generality:
Let $X$ be a Riemann surface, $Y$ its universal cover, and $\pi:Y\rightarrow X$ the universal covering map. Let $\Sigma=\big\{\{U_i\},\{\varphi_j\}\big\}$ be the complex structure on $X$. Then, $\Sigma$ may be lifted to $Y$ by way of $\pi$ to induce a complex structure $\Sigma'=\big\{\{\pi^{-1}|_{U_i}\},\{\psi_j\equiv \varphi_j\circ\pi\}\big\}$ on $Y$.
Let $\mathscr{U}\subseteq Y$ open, then $\varphi\circ\pi\equiv \psi$ on $\mathscr{U}$, which is to say that the coordinate neighborhoods have been chosen so that $\pi\equiv id_{\mathbb{D}}$ locally. In this case, $\pi$ is certainly a local biholomorphism.
Let $\pi'$ be a covering induced by some distinct complex structure $\Sigma^\ast$ on $Y$, and take $\mathscr{U}$ as before. If it holds for each $\mathscr{U}$ that there is some biholomorphism $f$ so that $f\circ\varphi\circ\pi'\equiv \psi$ on $\mathscr{U}$, then $\pi'$ is a local biholomorphism.
This really just amounts to saying that any universal covering map $\pi$ which is locally biholomorphic must be intimately related with $\Sigma'$, the complex structure on $Y$ induced by $X$.