Hello,

I am afraid that my main question might be a bit too elementary, but still I ask :

In short, my question is "what is the version of Montel's theorem for a family of holomorphic maps from an open subset $U$ to a hyperbolic Riemann surface $X$ whose range does not lie in a single co-ordinate chart of X ?" Here is how I got to think about the version :

I was reading up the proof of jenkins-Strebel quadratic differential of prescribed heights $b_i$ on a Riemann surface $X$ from kurt Strebel's book " Quadratic Differentials ", Chapter VI, section 21, page 108. Let us, for simplicity, use the version of the theorem in the case of one single Jordan curve $\gamma$. So, in the beginning of the proof, they consider a sequence of ring domains $R_n \subset X $ of homotopy type $\gamma $ with moduli $M_n$ such that $M_n \to M = sup_{n \ge 1 }M_n, 0 < M < \infty $. Then they consider a sequence of conformal maps $g_n : A_n \to R_n $, where $ A_n= { z \in \mathbb{C} : r_n < |z| < 1 } $, where $r_n \to 0 $.

Then they claim that : $ g_n : A_n \to R_n\subset X $ form a normal family. I understand that somehow they are trying to use Montels theorem from complex analysis : a family of holomorphic functions on a domain $U$ in $\mathbb{C}$ which omits two points is normal. But how exactly does this theorem apply to our present case, where the codomain of $g_n$ is a Riemann surface, not a subset of $\mathbb{C}$, so that we can NOT even talk about the absolute value symbol $ | g_n - g_m | $ , so what is the meaning of uniform convergence of a family of holomorphic maps which take values into a Riemann surface $X$ ? And what is the meaning of normality ?

I was thinking of ideas like lifting the maps $g_n $ first to the universal cover $D$ of $X$ with the normalization $g_n(p) =0 \forall n $ for some fixed $p \in A_1 $. Then we can apply Momtels's theorem to the lifted maps since all of them are missing three points, the co-domain being $D$, but then I needed to prove that the projection map $\pi: D\to X $ sends normal family to the normal family, which I was unable to prove.

This is my main question. A detailed explanations would be appreciated ! Thanks !