Holomorphic dependence for the Beltrami equation in Sobolev spaces

2024/10/08 by Emam, Christian El, Sagman, Nathaniel
#30C62 35J46 53C15 #Analysis of PDEs (math.AP) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2410.06175

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,∞_loc(Ω) of an open subset Ω⊂ \mathbb C, the canonical solutions to the Beltrami equation vary holomorphically in W^l+1,p_loc(Ω) 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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