Holomorphic convexity and Carleman approximation by entire functions on Stein manifolds
Let \(M\) be a totally real submanifold of a Stein manifold \(X\). One says that \(M\) admits a \(C^k\) Carleman approximation if for every \(f\in C^k(M)\) and every strictly positive \(\varepsilon\in C(M)\) there exists \(g\in O(X)\) such that \(|g-f|_{k,x}< \varepsilon(x)\), where \(|g-f|_{k,x}\) is the distance between the \(k\)-jets of \(g\) and \(f\) at \(x\). Take a normal exhaustion \(\{M^k\}\) of \(M\) and define the holomorphically convex hull of \(M\) as \[ \widehat M= \bigcup^\infty_{k=1}\widehat M^k, \] where \(\widehat M^k\) is the holomorphically convex hull of \(\widehat M^k\). Set furthermore \[ h(M)= \overline{\widehat M\setminus M}. \] Let \(E=\{E^k\}\) be a normal exhaustion of \(X\). One says that \(M\) has bounded \(E\)-hulls if for every \(k\) the set \(h(E^k\cup M)\) is compact in \(X\). The main result of the paper is the following: A totally real submanifold \(M\) of class \(C^k\) of a Stein manifold \(X\) admits a \(C^k\) Carleman approximation if and only if \(M\) is holomorphically convex, i.e., \(\widehat M= M\), and has bounded \(E\)-hulls in \(X\).
scientific article; zbMATH DE number 5988208
scientific article