Periodic solutions of the sinh-Gordon equation and integrable systems

We study the space of periodic solutions of the elliptic sinh-Gordon equation by means of spectral data consisting of a Riemann surface Y and a divisor D. We show that the space Mgp of real periodic finite type solutions with fixed period p can be considered as a completely integrable system (Mgp,Ω,H2) with a symplectic form Ω and a series of commuting Hamiltonians (Hn)n∈N. In particular we relate the gradients of these Hamiltonians to the Jacobi fields (ωn)n∈N0 from the Pinkall-Sterling iteration. Moreover, a connection between the symplectic form Ω and Serre duality is established.