Sub-Hermitian Geometry and the Quantitative Newlander-Nirenberg Theorem

TL;DR

This paper develops a quantitative, intrinsic theory of sub-Hermitian geometry, providing necessary and sufficient conditions for complex vector fields to have specific regularity levels, extending classical results to a broader, more flexible setting.

Contribution

It introduces a new sub-Hermitian geometric framework with invariant conditions for regularity of complex structures, generalizing classical and real theories.

Findings

Provides intrinsic conditions for regularity of complex structures

Extends the quantitative sub-Riemannian theory to complex and elliptic structures

Establishes a unified setting for real and complex geometric theories

Abstract

Given a finite collection of $C^1$ complex vector fields on a $C^2$ manifold $M$ such that they and their complex conjugates span the complexified tangent space at every point, the classical Newlander-Nirenberg theorem gives conditions on the vector fields so that there is a complex structure on $M$ with respect to which the vector fields are $T^{0,1}$. In this paper, we give intrinsic, diffeomorphic invariant, necessary and sufficient conditions on the vector fields so that they have a desired level of regularity with respect to this complex structure (i.e., smooth, real analytic, or have Zygmund regularity of some finite order). By addressing this in a quantitative way we obtain a holomorphic analog of the quantitative theory of sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. We call this sub-Hermitian geometry. Moreover, we proceed more generally and obtain similar results for manifolds which have an associated formally integrable elliptic structure. This allows us to introduce a setting which generalizes both the real and complex theories.