The Riemann mapping theorem

(Sketch)

The proof maximizes \(\abs{f'(z_0)}\) over injective holomorphic maps \(f\colon\Omega\to\DD\) with \(f(z_0)=0\) and shows that a maximizer is surjective.

• Fix \(z_0\in \Omega\) and set \(\mathcal F = \ts{f\in \Hol(\Omega, \DD) \st f(z_0) = 0,\ f \text{ injective}}\).

• \(\mathcal F\) is nonempty. Choose \(a\in\CC\sm\Omega\). Since \(z-a\) does not vanish on the simply connected set \(\Omega\), it has a holomorphic square root \(g\) on \(\Omega\), and \(g\) is injective. By the open mapping theorem \(g(\Omega)\) contains a disc \(D_r(g(z_0))\), and \(g(\Omega)\) is disjoint from \(D_r(-g(z_0))\): if \(g(z)\in D_r(-g(z_0))\), then \(-g(z) = g(z')\) for some \(z'\in\Omega\), so \(z = z'\) after squaring and \(g(z) = 0\), which contradicts \(a\notin\Omega\). Then \(z\mapsto (r/2)/(g(z)+g(z_0))\) is injective and maps \(\Omega\) into \(\DD\), and composing with the Blaschke factor that sends its value at \(z_0\) to \(0\) gives an element of \(\mathcal F\).

• For \(w\in\DD\), the automorphisms \(h_w\) of \(\DD\) satisfy \[ \begin{aligned} h_w(z) &\coloneqq {z-w \over 1-\bar{w} z} \in \Hol(\DD) \\ h'_w(0) &= 1 - \abs{w}^2 \\ h'_w(w) &= {1\over 1 - \abs{w}^2} \end{aligned}.\]

• A non-surjective \(f\in\mathcal F\) is not a maximizer. Let \(w\in\DD\sm f(\Omega)\); then \(w\neq 0\) and \(h_w\circ f\) is injective and nonvanishing on \(\Omega\), with value \(-w\) at \(z_0\). Let \(s\) be a holomorphic square root of \(h_w\circ f\), which is injective, and \(F\coloneqq h_{s(z_0)}\circ s\in\mathcal F\). Then \(\abs{s(z_0)}^2 = \abs w\) and \[ \abs{F'(z_0)} = \frac{\abs{s'(z_0)}}{1-\abs w} = \frac{(1-\abs w^2)\abs{f'(z_0)}}{2\sqrt{\abs w}\,(1-\abs w)} = \frac{1+\abs w}{2\sqrt{\abs w}}\abs{f'(z_0)} > \abs{f'(z_0)} .\]

• The family is bounded by \(1\), hence normal by Montel’s theorem.

• Set \(m\coloneqq \sup_{f\in \mathcal F} \abs{f'(z_0)}\). Then \(m>0\) because injective holomorphic maps have nonvanishing derivative, and \(m<\infty\) because, for \(\overline{D_R(z_0)}\subseteq\Omega\), the Cauchy estimate gives \[ \abs{f'(z_0)} \leq \max_{\abs{z-z_0} = R} { \abs{f(z)} \over R} \leq {1\over R} .\]

• Take \(\ts{f_k}\subseteq\mathcal F\) with \(\abs{f_k'(z_0)} \to m\), and use Montel to extract a locally uniformly convergent subsequence with limit \(h\).

• \(h\in\mathcal F\): \(h\) is holomorphic with \(h(z_0)=0\) and \(\abs{h'(z_0)} = m>0\), so it is nonconstant, maps into \(\DD\) by the maximum modulus principle, and is injective by Hurwitz’s theorem as a locally uniform limit of injective maps.

• \(h\) is surjective, since a non-surjective element of \(\mathcal F\) is not a maximizer.

• \(f'\neq 0\) on \(U\): if \(f'(z_0)=0\), then \(f - f(z_0)\) has a zero of order \(k\geq 2\) at \(z_0\), and \(f'\) has no other zeros near \(z_0\). For a small circle \(C\) about \(z_0\) and \(w\) with \(0<\abs w<\min_C\abs{f-f(z_0)}\), Rouché’s theorem shows that \(f - f(z_0) - w\) has \(k\) zeros inside \(C\), all simple since \(f'\neq 0\) away from \(z_0\), contradicting injectivity.

• By the open mapping theorem \(f(U)\) is open and \(f\) maps open sets to open sets, so \(g \coloneqq \inverseof{f}\colon f(U)\to U\) is continuous.

• For \(w, w_0\in f(U)\) with \(w\neq w_0\), write \(z = g(w)\) and \(z_0 = g(w_0)\), so \(z\neq z_0\), and \(z\to z_0\) as \(w\to w_0\) by continuity of \(g\): \[ \frac{g(w)-g\left(w_{0}\right)}{w-w_{0}}=\frac{1}{\frac{w-w_{0}}{g(w)-g\left(w_{0}\right)}}=\frac{1}{\frac{f(z)-f\left(z_{0}\right)}{z-z_{0}}} \convergesto{z\to z_0} {1\over f'(z_0) } = {1\over f'(g(w_0))} .\]