Each copy of the real line in \(\mathbb{C}^2\) is removable (Q2739260)

A closed subset \(X\) in \({\mathbb C}^n\) is called removable if every holomorphic function defined in \({\mathbb C}^n\backslash X\) extends holomorphically through \(X\). NEWLINENEWLINENEWLINEThe purpose of this very short note is to show that every closed set \(X\subset {\mathbb C}^2\) which is homeomorphic to the real line \({\mathbb R}\) is removable. The problem answered by E. Santillan Zeron has been raised around 1995 after the works of Lupacciolu, Chirka and Stout. In fact, as a motivation, it was known that for arbitrary closed subsets \(X\subset {\mathbb C}^n\) of topological dimension \(\leq n-2\), the Dolbeault cohomology groups \({}^\sigma H_c^{n,n-1}(X)\) and \(H_c^{n,n}(X)\) both vanish, hence \(X\) is removable, but it was also well known that many closed sets of topological dimension \(\leq 2n-3\) are also removable provided one has a measure-theoretic control of their thickness. NEWLINENEWLINENEWLINELet \(q\in X\cong {\mathbb R}\) be an arbitrary point and let \(f\in{\mathcal O}({\mathbb C}^2\backslash X)\). Here the topological property that \(X\) is homeomorphic to a real line is strongly used by the author to construct a nonempty relatively compact domain \(D\subset \subset {\mathbb C}^2\) whose boundary \(\partial D\) meets \(X\) in exactly two points \(p_1\) and \(p_2\) such that \(\partial D\backslash \{p_1,p_2\}\) is \({\mathcal C}^2\)-smooth, such that \(x\in D\) and \({\mathbb C}^2\backslash \overline{D}\) is connected (this last property is the most delicate one). By a theorem of \textit{G. Lupacciolu} [Ark. Mat. 32, No. 2, 455-473 (1994; Zbl 0823.32004)], it follows that the restriction \(f|_{\partial D\backslash \{p_1,p_2\}}\) extends holomorphically to \(D\), hence \(q\) is removable. For topological copies of \({\mathbb R}^{n-1}\) in \({\mathbb C}^n\) with \(n\geq 3\), this clever idea does not seem to apply directly, since it is unknown whether topological copies of \(S^{n-2}\) are removable for boundaries in \({\mathbb C}^n\).