A Frobenius-Nirenberg theorem with parameter

TL;DR

This paper extends the Newlander-Nirenberg theorem to include parameters, providing a parametric version with sharp regularity and a related Frobenius theorem with mild regularity loss.

Contribution

It introduces a parametric extension of the Newlander-Nirenberg theorem and a version of Nirenberg's complex Frobenius theorem with improved regularity results.

Findings

Established a parametric version of the Newlander-Nirenberg theorem.

Proved a Frobenius theorem with mild regularity loss.

Provided sharp regularity results for the parametric theorem.

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.