Proper holomorphic mappings onto symmetric products of a Riemann surface (Q1796609)

Summary: We show that the structure of proper holomorphic maps between the \(n\)-fold symmetric products, \(n\geq 2\), of a pair of non-compact Riemann surfaces \(X\) and \(Y\), provided these are reasonably nice, is very rigid. Specifically, any such map is determined by a proper holomorphic map of \(X\) onto \(Y\). This extends existing results concerning bounded planar domains, and is a non-compact analogue of a phenomenon observed in symmetric products of compact Riemann surfaces. Along the way, we also provide a condition for the complete hyperbolicity of all \(n\)-fold symmetric products of a non-compact Riemann surface.