On noncompactness of the $\overline{\partial }$-Neumann problem on pseudoconvex domains in ${\mathbb{C}}^3$

Abstract

In this paper we deal with the following question: is it true that any bounded smooth pseudoconvex domain in ${\mathbb{C}}^n$ whose boundary contains a $q$-dimensional complex manifold $M$ necessarily has a noncompact $\overline{\partial }$-Neumann operator $N_q$ ($1\le q\le n-1$)? We prove that a smooth bounded pseudoconvex domain $\Omega \subseteq {\mathbb{C}}^3$ with a one-dimensional complex manifold $M$ in its boundary has a noncompact Neumann operator on $(0,1)$-forms, under the additional assumption that $b\Omega$ has finite regular D'Angelo $2$-type at a point of $M$, improving previous results of Fu, Şahutoğlu, and Straube.