CMC hierarchy I: Commuting symmetries and loop algebra

TL;DR

This paper extends the theory of constant mean curvature surfaces into a hierarchy of evolution equations using loop algebra and symmetries, revealing new conservation laws and a generalized phase space.

Contribution

It introduces a novel CMC hierarchy derived from higher-order symmetries and constructs a loop algebra framework for analyzing these surfaces.

Findings

Development of a CMC hierarchy via bi-Hamiltonian structures

Extension of conservation laws to the hierarchy

Introduction of a generalized phase space for CMC surfaces

Abstract

We propose an extension of the structure equation for constant mean curvature (CMC) surfaces in a three dimensional Riemannian space form to the associated CMC hierarchy of evolution equations by the higher-order commuting symmetries. Via the canonical formal Killing field, considered as an infinitely prolonged and loop algebra valued Gau\ss$\,$ map, the CMC hierarchy is obtained by the assembly of a pair of Adler-Kostant-Symes bi-Hamiltonian hierarchies to the original CMC system. The infinite sequence of higher-order conservation laws of the CMC system admits the corresponding extension, and we find a formula for the generating series of the representative 1-forms. We also introduce a class of generalized (complexified) CMC surfaces as the phase space of the CMC hierarchy.