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

We construct a bounded pseudoconvex complete Reinhardt domain $\mathrm{\Omega }$ with smooth boundary in ${\mathbb{C}}^3$ such that the $\bar{\mathrm{\partial }}$-Neumann operator $N_1$ is compact although $b\mathrm{\Omega }$ 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 $\bar{\mathrm{\partial }}$-Neumann operator in the negative.