[en] 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.
Disciplines :
Mathematics
Author, co-author :
EL EMAM, Christian ✱; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)
SAGMAN, Nathaniel ✱; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)
✱ These authors have contributed equally to this work.
Language :
English
Title :
Holomorphic dependence for the Beltrami equation in Sobolev spaces
