Genus one Scherk surfaces and their limits

Abstract

The singly periodic, genus one helicoid was conjectured to be the limit of a one parameter family of doubly periodic minimal surfaces referred to as Perturbed Genus One Scherk Surfaces. Using elliptic functions, we show such surfaces exist, solving a two-dimensional period problem by perturbing a one-dimensional problem. Using flat structures associated to these minimal surfaces, we verify the conjecture.