On $1/2$ estimate for global Newlander-Nirenberg theorem
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910071136452608 |
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| author | Shi, Ziming |
| author_facet | Shi, Ziming |
| contents | Given a formally integrable almost complex structure $X$ defined on the closure of a bounded domain $D \subset \mathbb C^n$, and provided that $X$ is sufficiently close to the standard complex structure, the global Newlander-Nirenberg problem asks whether there exists a global diffeomorphism defined on $\overline D$ that transforms $X$ into the standard complex structure, under certain geometric and regularity assumptions on $D$. In this paper we prove a quantitative result of this problem. Assuming $D$ is a strictly pseudoconvex domain in $\mathbb C^n$ with $C^2$ boundary, and that the almost structure $X$ is of the Hölder-Zygmund class $Λ^r(\overline D)$ for $r>\frac{3}{2}$, we prove the existence of a global diffeomorphism (independent of $r$) in the class $Λ^{r+\frac12-\varepsilon}(\overline D)$, for any $\varepsilon>0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_02215 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On $1/2$ estimate for global Newlander-Nirenberg theorem Shi, Ziming Complex Variables Differential Geometry 32T15, 32Q60, 32Q40 Given a formally integrable almost complex structure $X$ defined on the closure of a bounded domain $D \subset \mathbb C^n$, and provided that $X$ is sufficiently close to the standard complex structure, the global Newlander-Nirenberg problem asks whether there exists a global diffeomorphism defined on $\overline D$ that transforms $X$ into the standard complex structure, under certain geometric and regularity assumptions on $D$. In this paper we prove a quantitative result of this problem. Assuming $D$ is a strictly pseudoconvex domain in $\mathbb C^n$ with $C^2$ boundary, and that the almost structure $X$ is of the Hölder-Zygmund class $Λ^r(\overline D)$ for $r>\frac{3}{2}$, we prove the existence of a global diffeomorphism (independent of $r$) in the class $Λ^{r+\frac12-\varepsilon}(\overline D)$, for any $\varepsilon>0$. |
| title | On $1/2$ estimate for global Newlander-Nirenberg theorem |
| topic | Complex Variables Differential Geometry 32T15, 32Q60, 32Q40 |
| url | https://arxiv.org/abs/2301.02215 |