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.
