Global Newlander-Nirenberg theorem on domains with finite smooth boundary in complex manifolds
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913787885387776 |
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| author | Gong, Xianghong Shi, Ziming |
| author_facet | Gong, Xianghong Shi, Ziming |
| contents | 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 $s<r+1/2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_09334 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global Newlander-Nirenberg theorem on domains with finite smooth boundary in complex manifolds Gong, Xianghong Shi, Ziming Complex Variables Differential Geometry 32F10, 32A26, 32W05 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 $s<r+1/2$. |
| title | Global Newlander-Nirenberg theorem on domains with finite smooth boundary in complex manifolds |
| topic | Complex Variables Differential Geometry 32F10, 32A26, 32W05 |
| url | https://arxiv.org/abs/2410.09334 |