Formal conserved quantities for isothermic surfaces

Abstract

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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Notes

  1. 1.

    Recall the isomorphism \(\bigwedge ^{2}\mathbb R ^{n+1,1}\cong o(\mathbb R ^{n+1,1})\) via \((u\wedge v)w=(u,w)v-(v,w)u\), for all \(u,v,w\in \mathbb R ^{n+1,1}\).

References

  1. 1.

    Bianchi, L.: Ricerche sulle superficie isoterme e sulla deformazione delle quadriche. Ann. di Mat. 11, 93–157 (1905)

    Article  Google Scholar 

  2. 2.

    Burstall, F.E.: Isothermic surfaces: conformal geometry, Clifford algebras and integrable systems. Integrable systems, geometry, and, topology, pp. 1–82. MR2222512 (2008b:53006) (2006)

  3. 3.

    Burstall, F.E., Ferus, D., Pedit, F., Pinkall, U.: Harmonic tori in symmetric spaces and commuting Hamiltonian systems on loop algebras. Ann. Math. (2) 138(1), 173–212 (1993). MR1230929 (94m:58057)

    Article  MATH  MathSciNet  Google Scholar 

  4. 4.

    Burstall, F.E., Santos, S.D.: Special isothermic surfaces of type \(d\). J. Lond. Math. Soc (2) 85(2), 571–591 (2012). MR2901079

    Article  MATH  MathSciNet  Google Scholar 

  5. 5.

    Burstall, F.E., Calderbank, D.: Conformal submanifold geometry IV-V (in preparation)

  6. 6.

    Burstall, F., Pedit, F., Pinkall, U.: Schwarzian derivatives and flows of surfaces. Differential geometry and integrable systems (Tokyo, 2000) pp. 39–61. MR1955628 (2004f:53010) (2002)

  7. 7.

    Burstall, F.E., Donaldson, N.M., Pedit, F., Pinkall, U.: Isothermic submanifolds of symmetric \(R\)-spaces. J. Reine Angew. Math. 660, 191–243 (2011). MR2855825

    MATH  MathSciNet  Google Scholar 

  8. 8.

    Calapso, P.: Sulle superficie a linee di curvatura isoterme. Rendiconti Circolo Matematico di Palermo 17, 275–286 (1903)

    Article  MATH  Google Scholar 

  9. 9.

    Calapso, P.: Sulle trasformazioni delle superficie isoterme. Ann. di Mat. 24, 11–48 (1915)

    Article  Google Scholar 

  10. 10.

    Cieśliński, J., Goldstein, P., Sym, A.: Isothermic surfaces in E \(^3\) as soliton surfaces. Phys. Lett. A 205(1), 37–43 (1995). MR1352426 (96g:53005)

    Google Scholar 

  11. 11.

    Darboux, G.: Sur une classe de surfaces isothermiques liées à la déformations des surfaces du second degré. C.R. Acad. Sci. Paris 128, 1483–1487 (1899)

    Google Scholar 

  12. 12.

    Darboux, G.: Sur les surfaces isothermiques. Ann. Sci. École Norm. Sup. (3) 16, 491–508 (1899). MR1508975

  13. 13.

    Hertrich-Jeromin, U.: Introduction to Möbius differential geometry, London Mathematical Society Lecture Note Series, vol. 300. Cambridge University Press, Cambridge (2003). MR2004958 (2004g:53001)

    Google Scholar 

  14. 14.

    Musso, E., Nicolodi, L.: Special isothermic surfaces and solitons. Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000), pp. 129–148. MR1871005 (2003a:53017) (2001)

  15. 15.

    Santos, S.D.: Special isothermic surfaces. Ph.D. Thesis, University of Bath (2008)

  16. 16.

    Schief, W.K.: Isothermic surfaces in spaces of arbitrary dimension: integrability, discretization, and Bäcklund transformations—a discrete Calapso equation. Stud. Appl. Math. 106(1), 85–137 (2001). MR1805487 (2002k:37140)

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to S. D. Santos.

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Burstall, F.E., Santos, S.D. Formal conserved quantities for isothermic surfaces. Geom Dedicata 172, 191–205 (2014). https://doi.org/10.1007/s10711-013-9915-5

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Keywords

  • Special isothermic surfaces
  • Polynomial and formal conserved quantities

Mathematics Subject Classification

  • 53A30
  • 53A05