Period quotient maps of meromorphic 1-forms and minimal surfaces on Tori

Consider on a complex 1-dimensional torus T_λ an abelian differential of the second kind α_λ. Assign to each λ the period quotient of α_λ for two independent cycles on T_λ. For appropriate choices of α_λ, the image of the modular sphere under this map is simple to describe, extending the classical case of holomorphic α_λ Using this information for different α_λ simultaneously, we discuss applications to existence and uniqueness questions for the Chen-Gackstatter surface, the Costa surface, and the translation invariant periodic helicoid with handles. In particular, there is now a conceptual and essentially non-computational proof for the uniqueness of the Chen-Gackstatter surface.