This book develops a spectral theory for the integrable system of 2-dimensional, simply periodic, complex-valued solutions
u of the sinh-Gordon equation. Such solutions (if real-valued) correspond to certain constant mean curvature surfaces in
Euclidean 3-space. Spectral data for such solutions are defined (following ideas of 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 is solved along a line, i.e. the solution u is reconstructed on a line from the spectral data. Finally,
a Jacobi variety and Abel map for the spectral curve are constructed and used to describe the change of the spectral data
under translation of the solution u. The book's primary audience will be research mathematicians interested in the theory
of infinite-dimensional integrable systems, or in the geometry of constant mean curvature surfaces.