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Surfaces on Lie groups, on Lie algebras, and their integrability

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Abstract

It is shown that the problem of the immersion of a 2-dimensional surface into a 3-dimensional Euclidean space, as well as then-dimensional generalization of this problem, is related to the problem of studying surfaces in Lie groups and surfaces in Lie algebras. A particular case of the general formalism presented here implies that any surface can be characterized in terms of 2×2 matrices using an arbitrary parametrization. It is also shown that this generality of parametrization is useful for studying integrable surfaces, i.e. surfaces described by integrable equations. In particular starting from a suitable Lax pair (i.e. a suitable integrable equation), it is possible to construct explicitly large classes of integrable surfaces.

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References

  1. Hopf, H.: Differential Geometry in the Large. Lect. Notes Math., Vol.1000, Berlin, Heidelberg, New York: Springer, 1983

    Google Scholar 

  2. Wente, H. C.: Counterexample to a Conjecture of H. Hopf. Pac. J. Math.121, 193–243 (1986)

    Google Scholar 

  3. Abresch, U.: Constant Mean Curvature Tori in Terms of Elliptic Functions. J. Reine Angew. Math.394, 169–192 (1987)

    Google Scholar 

  4. Spruck, J.: The Elliptic Sinh-Gordon Equation and the Construction of Toroidal Soap Bubbles. In: Hildebrandt, S., Kinderlehrer, D., Miranda, M. (eds.) Calculus of Variations and Partial Differential Equations. Proceedings, Tronto 1986, Lect. Notes Math., Vol.1340, Berlin, Heidelberg, New York: Springer, 1988, pp. 273–301

    Google Scholar 

  5. Walter, R.: Explicit Examples to the H-Problem of Heinz Hopf. Geom. Dedicate23, 187–213 (1987)

    Google Scholar 

  6. Wente, H.C.: Twisted Tori of Constant Mean Curvature in ℝ3. In: Tromba, A. (ed.) Seminar on New Results in Nonlinear Partial Differential Equations, Aspects Math. Vol.E10, Braunschweig Wiesbaden: Vieweg, 1987, pp. 1–36

    Google Scholar 

  7. Pinkall, U., Sterling, I.: On the Classification of Constant Mean Curvature Tori. Ann. Math.130, 407–451 (1989)

    Google Scholar 

  8. Bobenko, A.I.: Integrable Surfaces. Funkts. Anal. Prilozh.24 (3), 68–69 (1990)

    Google Scholar 

  9. Abresch, U.: Old and New Periodic Solutions of the Sinh-Gordon Equation. In: Tromba, A. (ed.) Seminar on New Results in Nonlinear Partial Differential Equations, Aspects Math. Vol.E10, Braunschweig Wiesbaden: Vieweg, 1987, pp. 37–73

    Google Scholar 

  10. Hitchin, N.S.: Harmonic Maps from a 2-Torus to the 3-Sphere. J. Diff. Geom.31, 627–710 (1990)

    Google Scholar 

  11. Bobenko, A.I.: Math. Ann.290, 209–245 (1991)

    Article  Google Scholar 

  12. Levi, D., Sym, A.: Phys. Lett.149A, 381–387 (1990)

    Google Scholar 

  13. Bobenko, A.I.: Surfaces in Terms of 2 by 2 Matrices, Old and New Integrable Cases. In: Harmonic Maps and Integrable Systems, Fordy, A.P., Wood, J.C. (eds.), Aspects of Mathematics, Vieweg, 1994

  14. Novikov, S.P.: (ed.) Theory of Solitons. New York: Consultants Bureau, 1984

    Google Scholar 

  15. Ablowitz, M.J., Segur, H.: Solitons and the Inverse Scattering Transformation. SIAM Stud. Appl. Mat.4, Philadelphia: SIAM, 1981

    Google Scholar 

  16. Newell, A.C.: Solitons in Mathematics and Physics, CBMS-NSF Regional Conf. Ser. in Appl. Math.48, Philadelphia: SIAM, 1985

    Google Scholar 

  17. Faddeev, L.D., Takhtajan, V.E.: Hamiltonian Methods in the Theory of Solitons. Berlin, Heidelberg, New York: Springer, 1986

    Google Scholar 

  18. Fokas, A.S., Zakharov, V.E.: (eds.), Important Developments in Soliton Theory. Berlin, Heidelberg, New York: Springer, 1993

    Google Scholar 

  19. Lax, P.: Comm. Pure. Appl. Math.21, 467 (1968)

    Google Scholar 

  20. Sym, A.: Soliton Surfaces and Their Applications (Soliton Geometry from Spectral Problems). In: Geometrical Aspects of the Einstein Equations and Integrable Systems, Lect. Notes Phys., Vol.239, Berlin, Heidelberg, New York: Springer, 1985, pp. 154–231

    Google Scholar 

  21. Olver, P.J.: Applications of Lie Groups to Differential Equations. Berlin, Heidelberg, New York: Springer-Verlag, 1986

    Google Scholar 

  22. Cartan, E.: La Theorie des Groups Finis et Continus et la Geometrie Differentielle Traitees par la Methode du Repere Mobile. Editions Jaeques Gabay, 1992

  23. Griffiths, P.: On Cartan's method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry. Duke Math. J. (1974)

  24. Fokas, A.S., Gel'fand, I.M., Liu, Q.M.: The motion of curves, of surfaces, and their integrability. Preprint, 1995

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Authors and Affiliations

  1. Department of Mathematical Sciences, Loughborough University of Technology, LE11 3TU, Loughborough, UK

    A. S. Fokas

  2. Institute for Nonlinear Studies, Clarkson University, 13699-5815, Potsdam, New York, USA

    A. S. Fokas

  3. Department of Mathematics, Rutgers University, 08903, New Brunswick, NJ, USA

    I. M. Gelfand

Authors
  1. A. S. Fokas
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  2. I. M. Gelfand
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Communicated by A. Jaffe

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Fokas, A.S., Gelfand, I.M. Surfaces on Lie groups, on Lie algebras, and their integrability. Commun.Math. Phys. 177, 203–220 (1996). https://doi.org/10.1007/BF02102436

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  • DOI: https://doi.org/10.1007/BF02102436

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