Measurable Riemann mapping theorem

Remarks

When $\mu$ has compact support, one looks for $f$ of the form $f(z) = z + \frac{1}{\pi}\int_\C \frac{h(w)}{z - w}\, dA(w)$, where $h = f_{\bar z}$. The Beltrami equation then becomes $h = \mu + \mu S h$, where $S$ is the Beurling–Ahlfors transform. The operator $S$ is an isometry of $L^2$, and its norm on $L^p$ is close to $1$ for $p$ close to $2$. So the equation has a solution given by a Neumann series in $L^p$ for some $p > 2$. The general case follows by composing solutions. Uniqueness holds because two normalized solutions differ by a $1$-quasiconformal, hence conformal, self-map of $\C$ fixing $0$ and $1$, which is the identity. The theorem also gives analytic dependence of $f$ on $\mu$, a fact that is central in Teichmüller theory and holomorphic dynamics. The condition $\norm{\mu}_\infty < 1$ cannot be weakened to $\abs{\mu} < 1$ almost everywhere: for some such $\mu$ no homeomorphic solution exists.

Going deeper

For a student

The theorem is the main existence result for quasiconformal mappings. It turns questions about conformal structures into questions about bounded measurable functions, which can be changed freely. In complex dynamics it is used in quasiconformal surgery: one builds an invariant ellipse field and then straightens it to get a new holomorphic map. In Teichmüller theory it shows that Beltrami coefficients parametrize deformations of Riemann surfaces, and the holomorphic dependence on $\mu$ gives the complex structure of Teichmüller space. For $\mu = 0$ the solution is the identity, so the statement extends the uniqueness of conformal automorphisms of $\C$ fixing two points.