Global Newlander-Nirenberg theorem on domains with finite smooth boundary in complex manifolds

TL;DR

This paper proves a global Newlander-Nirenberg theorem for certain domains in complex manifolds with finite smooth boundary, showing that sufficiently close almost complex structures can be integrated into genuine complex structures via a diffeomorphism.

Contribution

It extends the Newlander-Nirenberg theorem to domains with finite smooth boundary in complex manifolds, using a homotopy formula and Nash-Moser iteration.

Findings

Constructed a homotopy formula for $ heta$-valued (0,1)-forms.

Applied Nash-Moser scheme to integrate almost complex structures.

Established conditions under which almost complex structures are integrable.

Abstract

Let $M$ be a relatively compact $C^2$ domain in a complex manifold $\mathcal M$ of dimension $n$. Assume that $H^{1}(M,\Theta)=0$ where $\Theta$ is the sheaf of germs of holomorphic tangent fields of $M$. Suppose that the Levi-form of the boundary of $M$ has at least 3 negative eigenvalues or at least $n-1$ positive eigenvalues pointwise. We first construct a homotopy formula for $\Theta$-valued $(0,1)$-forms on $\overline M$. We then apply a Nash-Moser iteration scheme to show that if a formally integrable almost complex structure of the H\"{o}lder-Zygmund class $\Lambda^r$ on $\overline M$ is sufficiently close to the complex structure on $ M$ in the H\"{o}lder-Zygmund norm $\Lambda^{r_0}(\overline M)$ for some $r_0>5/2$, then there is a diffeomorphism $F$ from $\overline M$ into $\mathcal M$ that transforms the almost complex structure into the complex structure on $F(M)$, where $F \in \Lambda^s(M)$ for all $s<r+1/2$.