On 1/2 estimate for global Newlander-Nirenberg theorem

2023/01/05 by Ziming Shi, Shi, Ziming · 1 citation
Mathematics · #32Q40 #32Q60 #32T15 #Advanced Mathematical Physics Problems #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2301.02215

openalex publication_date 2023/01/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a formally integrable almost complex structure X defined on the closure of a bounded domain D ⊂ \mathbb Cⁿ, and provided that X is sufficiently close to the standard complex structure, the global Newlander-Nirenberg problem asks whether there exists a global diffeomorphism defined on D that transforms X into the standard complex structure, under certain geometric and regularity assumptions on D. In this paper we prove a quantitative result of this problem. Assuming D is a strictly pseudoconvex domain in \mathbb Cⁿ with C² boundary, and that the almost structure X is of the Hölder-Zygmund class Λ^r( D) for r>(3)/(2), we prove the existence of a global diffeomorphism (independent of r) in the class Λ^r+\frac12-ε( D), for any ε>0.

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