2014/09/23 by Joe S. Wang, Wang, Joe S.
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Waves and Solitons #math.DG
paper · pdf · doi:10.48550/arxiv.1409.7024
31 pages. This replaces arXiv:1312.7169
arxiv created 2014/09/23 · openalex publication_date 2014/09/23 · arxiv updated 2014/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.