Global Newlander-Nirenberg theorem on domains with finite smooth boundary in complex manifolds
Xianghong Gong, Ziming Shi
TL;DR
This work proves a global Newlander–Nirenberg type theorem on domains with finite smooth boundary in complex manifolds under the cohomology and Levi-form hypotheses stated. It develops a global homotopy formula for $\Theta$-valued $(0,1)$-forms and implements a Nash–Moser iteration to produce a diffeomorphism that straightens a nearby formally integrable almost complex structure, gaining almost $1/2$ derivative in the process. The argument combines local homotopy formulas, Grauert bumping, and $\overline\partial$-Neumann theory, together with Hartogs extension to handle concave boundary regions, yielding a regularity conclusion $F\in\Λ^{s}$ for all $s<r+1/2$ when $A\in\Λ^{r}$ with $r>r_0>5/2$. The results extend global and boundary Newlander–Nirenberg phenomena to $a_q^+$ domains and provide stability of cohomology under small domain perturbations, with implications for complex-analytic embedding problems in geometric analysis.
Abstract
Let $M$ be a relatively compact $C^2$ domain in a complex manifold $\mathcal M$ of dimension $n$. Assume that $H^{1}(M,Θ)=0$ where $Θ$ 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 $Θ$-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 Hölder-Zygmund class $Λ^r$ on $\overline M$ is sufficiently close to the complex structure on $ M$ in the Hölder-Zygmund norm $Λ^{r_0}(\overline M)$ 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 $F \in Λ^s(M)$ for all $s<r+1/2$.