Removable singularities and Liouville-type property of analytic multivalued functions
The main results of the paper are the following two theorems. 1) Let \(S\) be a closed subset of an open set \(G\subset\mathbb{C}^ n\) and let \(Y\) be a Stein space. Suppose that \(K:G\backslash S\to F_ c(Y)\) is an analytic multivalued function \((F_ c(Y):=\{A\subset Y:A\) is a compact, \(A\neq\emptyset\})\). Then \(K\) extends analytically to \(G\) if one of the following conditions is satisfied. a) \(S=H\cap(G\backslash U)\), where \(H\) is an analytic set in \(G\) and \(U\) is an open subset of \(G\) such that \(U\) intersects any connected component of \(H\). b) \(H\) is of \((2n-2)\)-Hausdorff measure zero. c) \(S\) is pluripolar and \(K\) is locally compact, i.e. each point \(x\in G\) has an open neighbourhood \(U\) such that \(K(U\backslash S)\) is relatively compact. 2) Let \(D\) be a convex domain in \(\mathbb{C}^ n\). The following conditions are equivalent: (i) For every analytic multivalued function \(K:\mathbb{C}\to F_ e(D)\) the function \(\hat K(x):=\) the polynomially convex hull of \(K(x)\), \(x\in\mathbb{C}\), is constant. (ii) Every analytic multivalued function \(K:\Delta\backslash\{0\}\to F_ c(D)\) extends analytically to \(\Delta\), where \(\Delta\) denotes the unit disc. (iii) For arbitrary polar set \(S\subset\Delta\) every analytic multivalued function \(K:\Delta\backslash S\to F_ c(D)\) extends analytically to \(\Delta\).
scientific article; zbMATH DE number 202539