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 $W_{loc}^{l,\infty }(\Omega )$ of an open subset $\Omega \subset \mathbb{C}$, the canonical solutions to the Beltrami equation vary holomorphically in $W_{loc}^{l+1,p}(\Omega )$ 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.

arXiv comment16 pages. Corrected minor typos and made small corrections