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Analytic discs and compactness of the ∂̅-Neumann operator
Abstract
We construct a bounded pseudoconvex complete Reinhardt domain Ω with smooth boundary in ℂ3 such that the ∂̅-Neumann operator N1 is compact although bΩ contains an analytic disc and thus also fails Catlin's Property (P) and McNeal's Property (P̃). This example solves an open problem on compactness of the ∂̅-Neumann operator in the negative.
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BibTeX
@misc{wang2026analytic,
title = {{Analytic discs and compactness of the $\bar\partial$-Neumann operator}},
author = {Qianyun Wang and Yuan Yuan and Xu Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01561},
eprint = {2609.01561},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01561},
}
RIS
TY - UNPB
TI - Analytic discs and compactness of the ∂̅-Neumann operator
AU - Qianyun Wang
AU - Yuan Yuan
AU - Xu Zhang
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01561
UR - https://arxiv.org/abs/2609.01561
AB - We construct a bounded pseudoconvex complete Reinhardt domain Ω with smooth boundary in ℂ^3 such that the ∂̅-Neumann operator N_1 is compact although bΩ contains an analytic disc and thus also fails Catlin's Property (P) and McNeal's Property (P̃). This example solves an open problem on compactness of the ∂̅-Neumann operator in the negative.
ER -