Compactness of the ∂̄-Neumann operator versus analytic discs in the boundary

The problem asks whether a boundary analytic disc necessarily prevents compactness of the $\bar{\partial}$-Neumann operator $N_1$. A new construction claims a negative answer, including failure of Catlin’s Property $(P)$ and McNeal’s Property $(\widetilde P)$ despite compactness.

Known results

• The unrestricted question was described as open in 2017; noncompactness was proved in $\mathbb{C}^3$ under finite regular D’Angelo $2$-type.
• The answer is affirmative in $\mathbb{C}^2$; Fu and Straube proved it for bounded locally convexifiable domains.
• For bounded domains with bounded intrinsic geometry, compactness of $N_q$ is equivalent to absence of $q$-dimensional analytic varieties in the boundary (2021).
• In that geometric setting, compactness is also equivalent to McNeal’s condition $(P_q)$ (2021).

September 2026 claimed counterexample

On September 1, 2026, a report linked the preprint Analytic discs and compactness of the $\bar{\partial}$-Neumann operator, which claims compactness of $N_1$ alongside a boundary analytic disc. If correct, this would refute the general obstruction and separate compactness from $(P)$ and $(\widetilde P)$; the claim is unverified.

Current status (as of September 2026): A preprint claims the general question is settled negatively, but the counterexample and proof remain unverified; no independent verification or referee report was found.