Work

Holomorphic dependence for the Beltrami equation in Sobolev spaces

Year: 2026

Type: article

Abstract: 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.

Source: Proceedings of the American Mathematical Society

DOI: 10.1090/proc/17402

Authors: Christian El Emam, Nathaniel Sagman

Institution: University of Luxembourg

Language: English

Topic: Analytic and geometric function theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: European Commission, Fonds National de la Recherche Luxembourg

Awards: G4350604260, G4541110996