Extensions of bounded holomorphic functions on the tridisk (Q783781)

Let \(\Omega\subset\mathbb C^d\) be a bounded domain and let \(\emptyset\neq\mathcal V\subset\Omega\). Assume that \(\mathcal V\) is a relatively polynomial convex subset of \(\Omega\) (i.e., \(\overline{\mathcal V}\) is polynomially convex and \(\overline{\mathcal V}\cap\Omega=\mathcal V\)) and has the polynomial extension property with respect to \(\Omega\) (i.e., for every polynomial \(f\in\mathbb C[z_1,\dots,z_n]\) there exists a \(\varphi\in H^\infty(\Omega)\) such that \(\varphi=f\) on \(\mathcal V\) and \(\sup_\Omega|\varphi|=\sup_{\mathcal V}|f|\)). The authors study the problem whether \(\mathcal V\) is a holomorphic retract of \(\Omega\). It is known that the answer is positive in the case where e.g. \(\Omega=\mathbb D^2\) is the bidisk or \(\Omega\) is the Euclidean ball. The authors discuss the case where \(\Omega=\mathbb D^3\) and they prove the following two theorems: \begin{itemize} \item Assume that \(\dim\mathcal V=1\). If \(\mathcal V\) is algebraic or has polynomially convex projections, then \(\mathcal V\) is a retract of \(\mathbb D^3\). \item Assume that \(\dim\mathcal V=2\). Then either \(\mathcal V\) is a retract or there exist domains \(U_r\subset\mathbb D^2\) and a holomorphic functions \(h_r:U_r\longrightarrow\mathbb D\), \(j=1,2,3\), such that \(\mathcal V =\{(z_1, z_2, h_3(z_1, z_2)) : (z_1, z_2)\in U_3\}= \{(z_1, h_2(z_1, z_3), z_3) : (z_1 , z_3)\in U_2\}=\{(h_1(z_2, z_3), z_2, z_3) : (z_2, z_3)\in U_1\}\). \end{itemize}