Geometric Realizations of Cyclically Branched Coverings over Punctured Spheres

Abstract

In classical differential geometry, a central question has been whether abstract surfaces with given geometric features can be realized as surfaces in Euclidean space. Inspired by the rich theory of embedded triply periodic minimal surfaces, we seek examples of triply periodic polyhedral surfaces that have an identifiable conformal structure. In particular, we are interested in explicit cone metrics on compact Riemann surfaces that have a realization as the quotient of a triply periodic polyhedral surface. This is important as Riemann surfaces where one has equivalent descriptions are rare. We construct periodic surfaces using graph theory as an attempt to make Schoen's heuristic concept of a dual graph rigorous. We then apply the theory of cyclically branched coverings to identify the conformal type of such surfaces.