What question did this study set out to answer?

The aim is to establish holomorphic dependence of canonical solutions for the Beltrami equation in Sobolev spaces.

March 15, 2026Proceedings of the American Mathematical Society0 citations

Holomorphic dependence for the Beltrami equation in Sobolev spaces

CEChristian El EmamCollegio Carlo Alberto NSNathaniel SagmanUniversity of Luxembourg

Key Points

The aim is to establish holomorphic dependence of canonical solutions for the Beltrami equation in Sobolev spaces.
Analyzed families of Beltrami forms with L-infinity norms under certain conditions.
Determined variational properties in the Sobolev space W_loc^{l, ext{infinity}}(Ω).
Extended results from previous foundational work by Ahlfors and Bers.
Canonical solutions vary holomorphically in the Sobolev space W_loc^{l+1,p}(Ω) for a specific p dependent on η.
Established that Bers metrics exhibit holomorphic dependence on input data.

Abstract

We prove that, given a family of Beltrami forms on C C with L ∞ L^ norm at most η > 1 >1 and that live in and vary holomorphically in the Sobolev space W l o c l, ∞ (Ω) W ₋₎₂^l, () of an open subset Ω ⊂ C C, the canonical solutions to the Beltrami equation vary holomorphically in W l o c l + 1, p (Ω) W ₋₎₂^l+1, p (), for some p = p (η) > 2 p=p () >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.

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