Holomorphic dependence for the Beltrami equation in Sobolev spaces

We prove that, given a family of Beltrami forms on




C

\mathbb {C}



with




L

∞



L^\infty



norm at most





η


>
1

\eta >1



and that live in and vary holomorphically in the Sobolev space





W


l
o
c



l
,

∞




(

Ω


)

W_{\mathrm {loc}}^{l,\infty }(\Omega )



of an open subset





Ω



⊂



C


\Omega \subset \mathbb {C}



, the canonical solutions to the Beltrami equation vary holomorphically in





W


l
o
c



l
+
1
,
p


(

Ω


)

W_{\mathrm {loc}}^{l+1,p}(\Omega )



, for some




p
=
p
(

η


)
>
2

p=p(\eta )>2



. This extends a foundational result of Ahlfors and Bers (the case




l
=
0

l=0



). As an application, we deduce that Bers metrics depend holomorphically on their input data.