I am trying to understand the proof of theorem 27.9. that every simply connected riemannian surface is biholomorphic to complex projective space, to the complex numbers or to the unit ball.
In particular he says that for some Rungian domains $Y_0\subset Y_1\subset...$ (which is not relevant here I assume) I can find biholomorphic maps from $Y_n$ to the unit ball $E$. Now we choose a point $a\in Y_0$ with local coordinates (U,z).
He then claims that there is a map $f_n: Y_n \to E(r_n)$, (where $E(r_n)$ is the ball with radius $r_n>0$), s.t. $f_n(a)=0$ and $df_n/dz(a)=1$. But I do not see why there should be such a map. I assume he is using some kind of Lemma 27.7. here, but first you would need a map from $Y_n$ to some $E(R)$ which maps $a$ to $0$.
My idea was maybe that we first map $Y_n$ to the unit ball with a map $h_n$ and then consider $h_n(a)\in E$, which we can translate under some special kind of biholomorphic translation such that it becomes the centre of a new ball with Radius $R$?