The pluricomplex Poisson kernel for convex finite type domains
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915525156667392 |
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| author | Arosio, Leandro Bracci, Filippo Fiacchi, Matteo |
| author_facet | Arosio, Leandro Bracci, Filippo Fiacchi, Matteo |
| contents | Given a bounded convex domain $D\subset \mathbb C^n$ of finite D'Angelo type and a boundary point $ξ\in \partial D$, we prove that the homogeneous complex Monge-Ampère equation $(dd^cu)^n=0$ possesses a continuous strictly negative solution $Ω_ξ$ that vanishes on $\partial D\setminus \{ξ\}$ and has a simple pole at $ξ$. We establish that $Ω_ξ(z)$ equals (up to sign) the normal derivative at $ξ$ of the pluricomplex Green function $G_z$, and its sublevel sets are the horospheres centered at $ξ$. Moreover, $Ω_ξ$ satisfies a Phragmen-Lindelöf type-theorem and provides a reproducing formula for plurisubharmonic functions. Consequently, $Ω_ξ$ serves as a generalisation of the classical Poisson kernel of the unit disc. Our approach, based on metric methods and scaling techniques, allows our results to be applied to strongly convex domains with $C^2$-smooth boundaries as well. In the course of the proof, we also establish a novel estimate of the Kobayashi distance near boundary points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_26230 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The pluricomplex Poisson kernel for convex finite type domains Arosio, Leandro Bracci, Filippo Fiacchi, Matteo Complex Variables 32U35, 32A19, 32F18, 32A40 Given a bounded convex domain $D\subset \mathbb C^n$ of finite D'Angelo type and a boundary point $ξ\in \partial D$, we prove that the homogeneous complex Monge-Ampère equation $(dd^cu)^n=0$ possesses a continuous strictly negative solution $Ω_ξ$ that vanishes on $\partial D\setminus \{ξ\}$ and has a simple pole at $ξ$. We establish that $Ω_ξ(z)$ equals (up to sign) the normal derivative at $ξ$ of the pluricomplex Green function $G_z$, and its sublevel sets are the horospheres centered at $ξ$. Moreover, $Ω_ξ$ satisfies a Phragmen-Lindelöf type-theorem and provides a reproducing formula for plurisubharmonic functions. Consequently, $Ω_ξ$ serves as a generalisation of the classical Poisson kernel of the unit disc. Our approach, based on metric methods and scaling techniques, allows our results to be applied to strongly convex domains with $C^2$-smooth boundaries as well. In the course of the proof, we also establish a novel estimate of the Kobayashi distance near boundary points. |
| title | The pluricomplex Poisson kernel for convex finite type domains |
| topic | Complex Variables 32U35, 32A19, 32F18, 32A40 |
| url | https://arxiv.org/abs/2509.26230 |