A Spectral Theory for Simply Periodic Solutions of the Sinh Gordon Equation 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 books primary audience will be research mathematicians interested in the theory of infinite dimensional integrable systems or in the geometry of constant mean curvature surfaces