A New Invariant for Doubly Periodic Solutions of the Sinh-Gordon Equation

Spectral curves of constant mean curvature tori in Euclidean three-space are real hyperelliptic curves carrying two meromorphic differentials with integral periods. These transcendental period conditions are studied through Whitham deformations, which deform the curve while preserving the periods. We extend Whitham deformations to all real hyperelliptic curves of this type, by holding fixed the plane spanned by the periods, or this plane together with its complex structure. Under mild nondegeneracy conditions the resulting isoperiodic sets are manifolds of dimensions three and one, respectively. Along the one-dimensional isoperiodic sets, a generalization of the Willmore energy is a local parameter away from its critical points, to which the spectral curves of tori belong. Finally, using homology bases adapted to the branch points, we find discrete invariants of the isoperiodic sets of doubly periodic curves: a triple of integers and a refined conformal type.