Ahlfors–Bers theorem

Remarks

For $\mu$ with compact support, the proof writes the derivative $\partial_{\bar z}$ of the principal solution as a Neumann series in the Beurling transform $T$: $\mu + \mu T \mu + \mu T(\mu T \mu) + \cdots$. The series converges in $L^p$ for some $p > 2$, because $T$ is an isometry of $L^2$ and its norm on $L^p$ tends to $1$ as $p \to 2$. Each term depends holomorphically on $\mu$, and the convergence is locally uniform. The same argument gives real-analytic dependence when $\mu_t$ depends real-analytically on a real parameter.

Example: for $\mu_t = t \chi$ with $\chi$ the indicator of a disc and $\abs{t} < 1$, the normalized solutions $f^{\mu_t}$ form a holomorphic family of quasiconformal maps through $f^{0} = \id$.

The condition $\norm{\mu_t}_\infty < 1$ is needed: when $\norm{\mu}_\infty = 1$ the Beltrami equation is degenerate, and a homeomorphic solution need not exist.

Going deeper

For a student

This holomorphic dependence is what makes Teichmüller space a complex manifold: Beltrami coefficients give coordinates, and the solution maps vary holomorphically with them. It is also the tool behind holomorphic motions and quasiconformal surgery in complex dynamics, where one deforms a rational map by moving an invariant Beltrami coefficient in a holomorphic family.