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arXiv · 1409.7024

CMC hierarchy I: Commuting symmetries and loop algebra

Joe S. Wang
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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 Gauß$\,$ 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.

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Joe S. Wang. 2014-09-23. CMC hierarchy I: Commuting symmetries and loop algebra. https://arxiv.org/abs/1409.7024

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