Proper holomorphic mappings onto symmetric products of a Riemann surface

We show that the structure of proper holomorphic maps between the n -fold symmetric products, n≥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