Sharp Regularity for the Integrability of Elliptic Structures

2018/10/23 by Street, Brian
#2010: 58A30 (Primary) #53C15 (Secondary) #Analysis of PDEs (math.AP) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1810.10057

Abstract

As part of his celebrated Complex Frobenius Theorem, Nirenberg showed that given a smooth elliptic structure (on a smooth manifold), the manifold is locally diffeomorphic to an open subset of ℝ^r× ℂⁿ (for some r and n) in such a way that the structure is locally the span of (∂)/(∂ t₁),…, (∂)/(∂ t_r),\frac∂∂ z₁,…, \frac∂∂ z_n; where ℝ^r× ℂⁿ has coordinates (t₁,…, t_r, z₁,…, z_n). In this paper, we give optimal regularity for the coordinate charts which achieve this realization. Namely, if the manifold has Zygmund regularity of order s+2 and the structure has Zygmund regularity of order s+1 (for some s>0), then the coordinate chart may be taken to have Zygmund regularity of order s+2. We do this by generalizing Malgrange's proof of the Newlander-Nirenberg Theorem to this setting.

Related

• Sharp Hölder Regularity for Nirenberg's Complex Frobenius Theorem
• Sub-Hermitian Geometry and the Quantitative Newlander-Nirenberg Theorem
• A Frobenius-Nirenberg theorem with parameter
• Global Newlander-Nirenberg theorem on domains with finite smooth boundary in complex manifolds
• Partial regularity for harmonic maps, and related problems