Geometriae Dedicata

, Volume 172, Issue 1, pp 191–205 | Cite as

Formal conserved quantities for isothermic surfaces

  • F. E. Burstall
  • S. D. SantosEmail author
Original Paper


Isothermic surfaces in \(S^n\) are characterised by the existence of a pencil \(\nabla ^t\) of flat connections. Such a surface is special of type \(d\) if there is a family \(p(t)\) of \(\nabla ^t\)-parallel sections whose dependence on the spectral parameter \(t\) is polynomial of degree \(d\). We prove that any isothermic surface admits a family of \(\nabla ^t\)-parallel sections which is a formal Laurent series in \(t\). As an application, we give conformally invariant conditions for an isothermic surface in \(S^3\) to be special.


Special isothermic surfaces Polynomial and formal conserved quantities 

Mathematics Subject Classification

53A30 53A05 


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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of BathBathUK
  2. 2.Departamento de Matemática, Faculdade de Ciências, CMAFUniversidade de LisboaLisbonPortugal

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