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 $\partial \Omega \setminus F$ is $\mathcal C^{2}$-smooth and locally pseudoconvex at every point, then $\Omega$ 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.