Holomorphic dependence for the Beltrami equation in Sobolev spaces

TL;DR

This paper proves that solutions to the Beltrami equation depend holomorphically on parameters within Sobolev spaces, extending classical results and applying to the holomorphic variation of Bers metrics on surfaces.

Contribution

It extends Ahlfors and Bers' foundational result to Sobolev spaces with higher regularity, showing holomorphic dependence of solutions in these spaces.

Findings

Solutions vary holomorphically in Sobolev spaces for admissible p > 2

Extension of classical Beltrami equation results to Sobolev regularity

Holomorphic dependence of Bers metrics on input data

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.