Let M be a relatively compact domain with C2 boundary in a complex manifold ℳ of dimension n. Assume that H1(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 n − 1 positive eigenvalues pointwise. We first construct a homotopy formula for Θ-valued (0, 1)-forms on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\overline M}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mover> <mi>M</mi> <mo accent="false">¯</mo> </mover> </mrow> </math></EquationSource> </InlineEquation>. We then show that if a formally integrable almost complex structure of the Hölder-Zygmund class Λr on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\overline M}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mover> <mi>M</mi> <mo accent="false">¯</mo> </mover> </mrow> </math></EquationSource> </InlineEquation> is sufficiently close to the complex structure on M in the Hölder-Zygmund norm <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\Lambda}^{{r_{0}}}({\overline M})$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msup> <mrow> <mi mathvariant="normal">Λ</mi> </mrow> <mrow> <mrow> <msub> <mi>r</mi> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </mrow> </msup> <mo stretchy="false">(</mo> <mrow> <mover> <mi>M</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> for some r0 > 5/2, then there is a diffeomorphism F from <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\overline M}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mover> <mi>M</mi> <mo accent="false">¯</mo> </mover> </mrow> </math></EquationSource> </InlineEquation> into ℳ that transforms the almost complex structure into the complex structure on F(M), where F ∈ Λs(M) for all s < r + 1/2.

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

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发表日期：

2026年7月16日

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Xianghong Gong,
Ziming Shi

摘要

Let M be a relatively compact domain with C² boundary in a complex manifold ℳ of dimension n. Assume that H¹(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 n − 1 positive eigenvalues pointwise. We first construct a homotopy formula for Θ-valued (0, 1)-forms on ${\overline M}$ $\bar{M}$ . We then show that if a formally integrable almost complex structure of the Hölder-Zygmund class Λ^r on ${\overline M}$ $\bar{M}$ is sufficiently close to the complex structure on M in the Hölder-Zygmund norm ${\Lambda}^{{r_{0}}}({\overline M})$ $Λ^{r_{0}} (\bar{M})$ for some r₀ > 5/2, then there is a diffeomorphism F from ${\overline M}$ $\bar{M}$ into ℳ that transforms the almost complex structure into the complex structure on F(M), where F ∈ Λ^s(M) for all s < r + 1/2.