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

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

Nicolas Hasse, Ross Ogilvie, Martin U. Schmidt

math.DG Oct 29, 2026 · v1
formalization geometry analysis
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TL;DR
The paper's statements are formalized in Lean 4, conditional on an axiomatic description of path integrals.
Abstract
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.
Problem

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 hard to study directly. Whitham (isoperiodic) deformations, which deform the curve while preserving periods, had been used only within the closure of spectral curves of tori.

Approach

Whitham deformations are extended to all nonsingular real hyperelliptic curves of this type. The deformation holds fixed the plane spanned by the periods, viewed as a point in Gr(2,R^g), or that plane together with its complex structure, viewed as a point in CP^{g-1}. Homology bases adapted to the branch points are used to build invariants of the isoperiodic sets. The statements are formalized in Lean 4, assuming an axiomatic treatment of path integrals.

Results

Under mild nondegeneracy conditions the isoperiodic sets are manifolds of dimension three and one, respectively. Along the one-dimensional sets, a generalized Willmore energy serves as a local parameter away from its critical points, and the spectral curves of tori lie at those critical points. For doubly periodic curves, the isoperiodic sets carry discrete invariants: a triple of integers and a refined conformal type.