Removable sets for pseudoconvexity for weakly smooth boundaries

Abstract

We show that for bounded domains in ${\mathbb{C}}^n$ with ${\mathcal{C}}^{1,1}$ smooth boundary, if there is a closed set $F$ of $2n-1$-Lebesgue measure $0$ such that $\partial \hat{I}\text{c◯}\setminus F$ is ${\mathcal{C}}^2$-smooth and locally pseudoconvex at every point, then $\hat{I}\text{c◯}$ is globally pseudoconvex. Unlike in the globally ${\mathcal{C}}^2$-smooth case, the condition ``$F$ of (relative) empty interior'' is not enough to obtain such a result. We also give some results under peak-set type hypotheses, which in particular provide a new proof of an old result of Grauert and Remmert about removable sets for pseudoconvexity under minimal hypotheses of boundary regularity.

Many typos corrected with the help of the anonymous referee; to appear in Math. Zeitschrift