Formal conserved quantities for isothermic surfaces


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.

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Burstall, F.E., Santos, S.D. Formal conserved quantities for isothermic surfaces. Geom Dedicata 172, 191–205 (2014).

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  • Special isothermic surfaces
  • Polynomial and formal conserved quantities

Mathematics Subject Classification

  • 53A30
  • 53A05