Global regularity for the $\bar\partial$-Neumann problem on pseudoconvex manifolds
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911982341324800 |
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| author | Khanh, Tran Vu Raich, Andrew |
| author_facet | Khanh, Tran Vu Raich, Andrew |
| contents | We establish general sufficient conditions for exact (and global) regularity in the $\bar\partial$-Neumann problem on $(p,q)$-forms, $0 \leq p \leq n$ and $1\leq q \leq n$, on a pseudoconvex domain $Ω$ with smooth boundary $bΩ$ in an $n$-dimensional complex manifold $M$. Our hypotheses include two assumptions: 1) $M$ admits a function that is strictly plurisubharmonic acting on $(p_0,q_0)$-forms in a neighborhood of $bΩ$ for some fixed $0 \leq p_0 \leq n$, $1 \leq q_0 \leq n$, or $M$ is a Kähler metric whose holomorphic bisectional curvature acting $(p,q)$-forms is positive; and 2) there exists a family of vector fields $T_ε$ that are transverse to the boundary $bΩ$ and generate one forms, which when applied to $(p,q)$-forms, $0 \leq p \leq n$ and $q_0 \leq q \leq n$, satisfy a "weak form" of the compactness estimate.
We also provide examples and applications of our main theorems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_04512 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global regularity for the $\bar\partial$-Neumann problem on pseudoconvex manifolds Khanh, Tran Vu Raich, Andrew Complex Variables 32W05, 32T99, 35N15, 32A25 We establish general sufficient conditions for exact (and global) regularity in the $\bar\partial$-Neumann problem on $(p,q)$-forms, $0 \leq p \leq n$ and $1\leq q \leq n$, on a pseudoconvex domain $Ω$ with smooth boundary $bΩ$ in an $n$-dimensional complex manifold $M$. Our hypotheses include two assumptions: 1) $M$ admits a function that is strictly plurisubharmonic acting on $(p_0,q_0)$-forms in a neighborhood of $bΩ$ for some fixed $0 \leq p_0 \leq n$, $1 \leq q_0 \leq n$, or $M$ is a Kähler metric whose holomorphic bisectional curvature acting $(p,q)$-forms is positive; and 2) there exists a family of vector fields $T_ε$ that are transverse to the boundary $bΩ$ and generate one forms, which when applied to $(p,q)$-forms, $0 \leq p \leq n$ and $q_0 \leq q \leq n$, satisfy a "weak form" of the compactness estimate. We also provide examples and applications of our main theorems. |
| title | Global regularity for the $\bar\partial$-Neumann problem on pseudoconvex manifolds |
| topic | Complex Variables 32W05, 32T99, 35N15, 32A25 |
| url | https://arxiv.org/abs/2408.04512 |