Holomorphic dependence for the Beltrami equation in Sobolev spaces

Abstract

We prove that, given a path of Beltrami differentials on $\mathbb C$ that live in and vary holomorphically in the Sobolev space $W^{l,\infty}_{loc}(\Omega)$ of an open subset $\Omega\subset \mathbb C$, the canonical solutions to the Beltrami equation vary holomorphically in $W^{l+1,p}_{loc}(\Omega)$ for admissible $p > 2$. This extends a foundational result of Ahlfors and Bers (the case $l = 0$). As an application, we deduce that Bers metrics on surfaces depend holomorphically on their input data.