Sharp Regularity for the Integrability of Elliptic Structures

TL;DR

This paper establishes optimal regularity conditions for coordinate charts in elliptic structures, extending classical results by showing that Zygmund regularity is preserved at a higher order, generalizing the Newlander-Nirenberg theorem.

Contribution

It provides the best possible regularity results for coordinate charts in elliptic structures with Zygmund regularity, extending Malgrange's proof of the Newlander-Nirenberg theorem.

Findings

Optimal Zygmund regularity for coordinate charts achieved

Extension of Malgrange's proof to elliptic structures

Regularity of charts matches the structure's regularity level

Abstract

As part of his celebrated Complex Frobenius Theorem, Nirenberg showed that given a smooth elliptic structure (on a smooth manifold), the manifold is locally diffeomorphic to an open subset of $\mathbb{R}^r\times \mathbb{C}^n$ (for some $r$ and $n$) in such a way that the structure is locally the span of $\frac{\partial}{\partial t_1},\ldots, \frac{\partial}{\partial t_r},\frac{\partial}{\partial \overline{z}_1},\ldots, \frac{\partial}{\partial \overline{z}_n}$; where $\mathbb{R}^r\times \mathbb{C}^n$ has coordinates $(t_1,\ldots, t_r, z_1,\ldots, z_n)$. In this paper, we give optimal regularity for the coordinate charts which achieve this realization. Namely, if the manifold has Zygmund regularity of order $s+2$ and the structure has Zygmund regularity of order $s+1$ (for some $s>0$), then the coordinate chart may be taken to have Zygmund regularity of order $s+2$. We do this by generalizing Malgrange's proof of the Newlander-Nirenberg Theorem to this setting.