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arXiv · 2609.01561

Analytic discs and compactness of the $\bar\partial$-Neumann operator

Qianyun Wang·Yuan Yuan·Xu Zhang
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Abstract

We construct a bounded pseudoconvex complete Reinhardt domain $\Omega$ with smooth boundary in $\mathbb{C}^3$ such that the $\bar\partial$-Neumann operator $N_1$ is compact although $b\Omega$ contains an analytic disc and thus also fails Catlin's Property $(P)$ and McNeal's Property $(\tilde P)$. This example solves an open problem on compactness of the $\bar\partial$-Neumann operator in the negative.

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Qianyun Wang, Yuan Yuan, Xu Zhang. 2026-09-01. Analytic discs and compactness of the $\bar\partial$-Neumann operator. https://arxiv.org/abs/2609.01561

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