The Newlander-Nirenberg theorem for complex $b$-manifolds

Abstract

Melrose defined the b-tangent bundle of a smooth manifold M with boundary as the vector bundle whose sections are vector fields on M tangent to the boundary. Mendoza defined a complex b-manifold as a manifold with boundary together with an involutive splitting of the complexified b-tangent bundle into complex conjugate factors. We prove complex b-manifolds have a single local model depending only on dimension. This can be thought of as the Newlander-Nirenberg theorem for complex b-manifolds. Our proof uses Mendoza's result that complex b-manifolds have no "formal local invariants" and a singular coordinate change to leverage the classical Newlander-Nirenberg theorem and Catlin's generalization for complex manifolds with pseudoconvex boundary.

v1: 16 pages, 1 figure. v2: 24 pages, 2 figures. In v1, the proof of Theorem 5.1 hid a gap in the assertion "...there is no harm in assuming gamma satisfies the periodicity condition...". In v2 we repair this gap. A new section "Extending deformed structures across boundaries" is added. Also, the basic setup is changed from "manifold with distinguished hypersurface" to "manifold with boundary"