The Complex Frobenius Theorem for Rough Involutive Structures

TL;DR

This paper extends the complex Frobenius theorem to low-regularity structures, specifically Lipschitz Levi-flat CR-manifolds, using a refined Newlander-Nirenberg theorem for integrable almost complex structures.

Contribution

It develops a version of the complex Frobenius theorem applicable to Lipschitz involutive structures, advancing the understanding of integrability under minimal regularity.

Findings

Established a Frobenius theorem for Lipschitz involutive structures.

Extended the Newlander-Nirenberg theorem to minimal regularity settings.

Applied results to Levi-flat CR-manifolds with Lipschitz structures.

Abstract

We establish a version of the complex Frobenius theorem in the context of a complex subbundle S of the complexified tangent bundle of a manifold, having minimal regularity. If the subbundle S defines the structure of a Levi-flat CR-manifold, it suffices that S be Lipschitz for our results to apply. A principal tool in the analysis is a precise version of the Newlander-Nirenberg theorem with parameters, for integrable almost complex structures with minimal regularity, which builds on previous recent work of the authors.