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Holomorphic dependence for the Beltrami equation in Sobolev spaces
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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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Cited by 1 Pith paper
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Complex harmonic maps and rank 2 higher Teichm\"uller theory
Complex harmonic maps are used to prove that rank-2 Hitchin components carry a mapping-class-group-invariant pseudo-Kähler structure and a Bers-type simultaneous uniformization.
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