= Green-function exhaustion proof of disk uniformization
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Exhaust a simply connected <hyperbolic Riemann surface in potential theory> by compact bordered Jordan domains $R_n$. Their <Green functions on a Riemann surface> $G_n(\cdot,p)$ have logarithmic poles and zero boundary values. Exponentiating their harmonic conjugates produces proper degree-one maps $f_n:R_n\to\mathbb D$ with $f_n(p)=0$. If $G_n=-\log|\zeta|+c_n+o(1)$, normalize $f_n'(p)=e^{-c_n}>0$. Domain monotonicity and a global positive Green function bound $c_n$ increasingly from above. Thus <Montel theorem> gives a nonconstant limit $F$, which is injective by the <locally uniform limit of univalent functions> theorem. The <Schwarz lemma> makes $F'(p)$ extremal among normalized disk-valued maps. If $F$ omitted $a\ne0$, taking a global <holomorphic square root> of a disk automorphism applied to $F$, then renormalizing at $p$, would multiply its derivative by $(1+|a|)/(2\sqrt{|a|})>1$. Thus $F$ is onto and gives the desired <biholomorphism>.
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