Removable sets for holomorphic functions of several complex variables (Q583433)

The author characterizes removable sets for holomorphic functions in terms of their Hausdorff measures. In particular, he proves the following two general theorems: 1. Let \(\Lambda^{\alpha}\) denote the standard \(\alpha\)-dimensional Hausdorff measure in \({\mathbb{R}}^{2N}\). If E is a closed subset of a k- dimensional complex submanifold \(M\subset {\mathbb{C}}^ N\) such that \(\Lambda^{2k-2}(E)<\infty\) then \({\mathcal O}(M\setminus E)={\mathcal O}(M)\), i.e. E is removable. 2. Let B be the unit ball in \({\mathbb{C}}^ N\) and let \(\Lambda_ B^{\alpha}\) denote the \(\alpha\)-dimensional Hausdorff measure with respect to the distance generated by the Bergman metric in B. If E is a closed subset of B with \(\Lambda_ B^{2N-2}(E)<\infty\) then E is removable.