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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.20817 |
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| _version_ | 1866915567148990464 |
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| author | Nguyen, Quang Dieu Thomas, Pascal J. |
| author_facet | Nguyen, Quang Dieu Thomas, Pascal J. |
| contents | We show that for bounded domains in $\mathbb C^n$ with $\mathcal C^{1,1}$ smooth boundary, if there is a closed set $F$ of $2n-1$-Lebesgue measure $0$ such that $\partial Ω\setminus F$ is $\mathcal C^{2}$-smooth and locally pseudoconvex at every point, then $Ω$ is globally pseudoconvex. Unlike in the globally $\mathcal C^{2}$-smooth case, the condition ``$F$ of (relative) empty interior'' is not enough to obtain such a result. We also give some results under peak-set type hypotheses, which in particular provide a new proof of an old result of Grauert and Remmert about removable sets for pseudoconvexity under minimal hypotheses of boundary regularity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_20817 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Removable sets for pseudoconvexity for weakly smooth boundaries Nguyen, Quang Dieu Thomas, Pascal J. Complex Variables Analysis of PDEs 32T99, 32W50 We show that for bounded domains in $\mathbb C^n$ with $\mathcal C^{1,1}$ smooth boundary, if there is a closed set $F$ of $2n-1$-Lebesgue measure $0$ such that $\partial Ω\setminus F$ is $\mathcal C^{2}$-smooth and locally pseudoconvex at every point, then $Ω$ is globally pseudoconvex. Unlike in the globally $\mathcal C^{2}$-smooth case, the condition ``$F$ of (relative) empty interior'' is not enough to obtain such a result. We also give some results under peak-set type hypotheses, which in particular provide a new proof of an old result of Grauert and Remmert about removable sets for pseudoconvexity under minimal hypotheses of boundary regularity. |
| title | Removable sets for pseudoconvexity for weakly smooth boundaries |
| topic | Complex Variables Analysis of PDEs 32T99, 32W50 |
| url | https://arxiv.org/abs/2504.20817 |