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

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 Hölder-Zygmund class ${\Lambda }^r$ on $\overline{M}$ is sufficiently close to the complex structure on $M$ in the Hölder-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 $s<r+1/2$.

60 pages