Conservation laws for surfaces of constant mean curvature in 3-dimensional space forms

TL;DR

This paper explores the symmetries, conservation laws, and spectral properties of constant mean curvature surfaces in 3D space forms, revealing integrable structures and geometric interpretations with implications for surface classification.

Contribution

It introduces an enhanced prolongation model for CMC surfaces, establishes a Noether's theorem correspondence, and analyzes spectral curves and conservation laws in a unified integrable systems framework.

Findings

Infinite sequence of higher-order symmetries and conservation laws identified.

Spectral conservation laws correspond to secondary characteristic cohomology classes.

Spectral curves are characterized by the eigenvalues of monodromies, linking to surface classification.

Abstract

The exterior differential system for constant mean curvature (CMC) surfaces in a 3-dimensional space form is an elliptic Monge-Ampere system defined on the unit tangent bundle. We determine the infinite sequence of higher-order symmetries and conservation laws via an enhanced prolongation modelled on a loop algebra valued formal Killing field. As a consequence we establish Noether's theorem for the CMC system and there is a canonical isomorphism between the symmetries and conservation laws. A geometric interpretation of the $\mathbb{S}^{1}$-family of associate surfaces leads to an integrable extension for a non-local symmetry called spectral symmetry. We show that the corresponding spectral conservation law exists as a secondary characteristic cohomology class. For a compact linear finite type CMC surface of arbitrary genus, we observe that the monodromies of the associated flat $\mathfrak{sl}(2,\mathbb{C})$-connection commute with each other. It follows that a single spectral curve is defined as the completion of the set of eigenvalues of the entire monodromies. This agrees with the recent result that a compact high genus linear finite type CMC surface necessarily factors through a branched covering of a torus. We introduce a sequence of Abel-Jacobi maps defined by the periods of conservation laws. For the class of deformations of CMC surfaces which scale the Hopf differential by a real parameter, we compute the first order truncated Picard-Fuchs equation. The resulting formulae for the first few terms exhibit a similarity with the Griffiths transversality theorem for variation of Hodge structures.