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PreprintCC BY 4.0arXiv

On the pluricomplex Poisson kernel and the associated complex Monge--Ampère equation

  • Xieping Wang
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Abstract

We give a geometric characterization of the pluricomplex Poisson kernel of bounded strongly linearly convex domains in ℂn+1 in terms of their foliation by all complex geodesic discs whose closure contains a fixed boundary point, thus confirming a conjecture posed by Bracci et al. in 2009, even in greater generality. The proof relies crucially on our previous results obtained in a series of two papers. We also revisit the homogeneous complex Monge--Ampère equation associated with the pluricomplex Poisson kernel and show that its solutions are far from unique in general by constructing a family of pairwise nonproportional continuous solutions on the unit ball in ℂn+1. This sharply contrasts with the uniqueness of solutions to the corresponding equation for the pluricomplex Green function.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026pluricomplex,
  title = {{On the pluricomplex Poisson kernel and the associated complex Monge--Ampère equation}},
  author = {Xieping Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.15572},
  eprint = {2609.15572},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.15572},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the pluricomplex Poisson kernel and the associated complex Monge--Ampère equation
AU  - Xieping Wang
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.15572
UR  - https://arxiv.org/abs/2609.15572
AB  - We give a geometric characterization of the pluricomplex Poisson kernel of bounded strongly linearly convex domains in ℂ^n+1 in terms of their foliation by all complex geodesic discs whose closure contains a fixed boundary point, thus confirming a conjecture posed by Bracci et al. in 2009, even in greater generality. The proof relies crucially on our previous results obtained in a series of two papers. We also revisit the homogeneous complex Monge--Ampère equation associated with the pluricomplex Poisson kernel and show that its solutions are far from unique in general by constructing a family of pairwise nonproportional continuous solutions on the unit ball in ℂ^n+1. This sharply contrasts with the uniqueness of solutions to the corresponding equation for the pluricomplex Green function.
ER  -