  This note summarizes results that were obtained by the author in his
habilitation thesis (arXiv:1607.08792) concerning the development of a spectral
theory for simply periodic, 2-dimensional, complex-valued solutions of the
sinh-Gordon equation. Spectral data for such solutions are defined for periodic
Cauchy data on a line (following Hitchin and Bobenko) and the space of spectral
data is described by an asymptotic characterization. Using methods of
asymptotic estimates, the inverse problem for the spectral data of such Cauchy
data is answered. Finally a Jacobi variety for the spectral curve is
constructed, and this is used to study the asymptotic behavior of the spectral
data corresponding to actual simply periodic solutions of the sinh-Gordon
equation on strips of positive height.
