2020/05/15 by Chun Gan, Gan, Chun, Xianghong Gong +1 · 1 citation
Mathematics · #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2005.07679
openalex publication_date 2020/05/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Newlander-Nirenberg theorem says that a formally integrable complex structure is locally equivalent to the standard complex structure in the complex Euclidean space. In this paper, we consider two natural generalizations of the Newlander-Nirenberg theorem under the presence of a C2 strictly pseudoconvex boundary. When a given formally integrable complex structure X is defined on the closure of a bounded strictly pseudoconvex domain with C2 boundary D⊂ ℂn, we show the existence of global holomorphic coordinate systems defined on D that transform X into the standard complex structure provided that X is sufficiently close to the standard complex structure. Moreover, we show that such closeness is stable under a small C2 perturbation of ∂ D. As a consequence, when a given formally integrable complex structure is defined on a one-sided neighborhood of some point in a C2 real hypersurface M⊂ ℂn, we prove the existence of local one-sided holomorphic coordinate systems provided that M is strictly pseudoconvex with respect to the given complex structure. We also obtain results when the structures are finite smooth.