Removable sets for pseudoconvexity for weakly smooth boundaries

par Pascal Thomas (IMT - Université de Toulouse)

lundi 24 nov. 2025, 14:00 → 15:00 Europe/Paris

Amphi Schwartz

The natural domains of existence of holomorphic functions in several variables are characterized by pseudoconvexity, which turns out to be a local condition on their boundaries. In the case where said boundary is $\mathcal C^{2}$-smooth, this can be understood as a sort of infinitesimal
plurisubharmonicity along the complex tangential directions, and written explicitly in terms of the Levi form of the defining function of the domain, which produces a non-linear second order partial differential inequation on the defining function.

We show that for bounded domains in $\C^n$ with $\mathcal C^{1,1}$ smooth boundary, if there is a closed set $F$ of $2n-1$-Lebesgue measure $0$ such that $\partial \Om \setminus F$ 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.