A Frobenius-Nirenberg theorem with parameter

2016/11/12 by Xianghong Gong, Gong, Xianghong · 2 citations
Mathematics · #32Q60 #32V05 #52C12 #Advanced Operator Algebra Research #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1611.03939

openalex publication_date 2016/11/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Newlander-Nirenberg theorem says that a formally integrable complex structure is locally equivalent to the complex structure in the complex Euclidean space. We will show two results about the Newlander-Nirenberg theorem with parameter. The first extends the Newlander-Nirenberg theorem to a parametric version, and its proof yields a sharp regularity result as Webster's proof for the Newlander-Nirenberg theorem. The second concerns a version of Nirenberg's complex Frobenius theorem and its proof yields a result with a mild loss of regularity.

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• The Oka principle for tame families of Stein manifolds
• Smooth equivalence of deformations of domains in complex euclidean spaces

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