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arXiv · 2410.06175

Holomorphic dependence for the Beltrami equation in Sobolev spaces

Christian El Emam·Nathaniel Sagman
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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.

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Christian El Emam, Nathaniel Sagman. 2024-10-08. Holomorphic dependence for the Beltrami equation in Sobolev spaces. https://arxiv.org/abs/2410.06175

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