Skip to main content
SpringerLink
Log in

Totally real minimal tori in

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract.

In this paper we show that all totally real minimal tori in correspond to doubly periodic finite gap solutions of the Tzitzéica equation Using the results on the Tzitzéica equation in integrable systems theory, we describe explicitly these tori by Prym-theta functions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abresch, U.: Constant mean curvature tori in terms of elliptic functions. J. Reine Angew. Math. 374, 169–192 (1987)

    MathSciNet  MATH  Google Scholar 

  2. Belokolos, E.D., Bobenko, A.I., Enol’skii, V.Z., Its, A.R., Matveev, V.M.: Algebro-geometric approach to nonlinear integrable equations. Springer Verlag, 1994

  3. Bobenko, A.I.: All constant mean curvature tori in R3, S3, H3 in terms of theta- functions. Math. Ann. 290, 209–245 (1991)

    MathSciNet  MATH  Google Scholar 

  4. Bobenko, A.I.: Exploring surfaces through methods from the theory of integrable systems. Lecture on Bonnet Problem. SFB288 Preprint 403 (1999)

  5. Bolton, J., Pedit, F., Woodward, L.M.: Minimal surfaces and the affine Toda field modle. J. Reine Angew. Math. 459, 119–150 (1995)

    MathSciNet  MATH  Google Scholar 

  6. Bolton, J., Woodward, L.M.: The affine Toda equations and minimal surfaces. In: Integrable Systems and Harmonic maps, A.P. Fordy, J.C. Wood, (eds.), Aspects of Mathematics E23, Vieweg, 1994

  7. Bullough, R.K., Dodd, R.K.: Polynomial conserved densities for the sine-Gordon equations. Proc. Roy. Soc. London Ser. A. 352, 481–503 (1977)

    Google Scholar 

  8. Burstall, F.E.: Harmonic tori in spheres and complex projective spaces. J. Reine Angew. Math. 469, 149–177 (1995)

    MathSciNet  MATH  Google Scholar 

  9. Burstall, F.E., Ferus, D., Pedit, F., Pincall, U.: Harmonic tori in symmetric spaces and commuting Hamiltonian systems on loop algebras. Ann. Math. 138, 173–212 (1993)

    MathSciNet  MATH  Google Scholar 

  10. Burstall, F.E., Pedit, F.: Harmonic maps via Adler-Kostant-Symes theory. In: Harmonic Maps and Integrable Systems, A.P. Fordy, J.C. Wood, (eds.), Aspects of Mathematics E23, Vieweg, 1994

  11. Cherdantsev, Yu.I., Sharipov, R.A.: Finite-gap solutions of the Bullough-Dodd-Zhiber-Shabat equation. Theor. Math. Phys. 82(1), 108–111 (1990)

    MATH  Google Scholar 

  12. Castro, I., Urbano, F.: New examples of minimal Lagrangian tori in the complex projective plane. Manuscripta Math. 85, 265–281 (1994)

    MathSciNet  MATH  Google Scholar 

  13. Dubrovin, B.A.: Theta-functions and non-linear equations. Uspekhi Mat. Nauk 36(2), 11–80 (1981); Russian Math. Surveys 36(2), 11–92 (1981)

    MATH  Google Scholar 

  14. Eells, J., Wood, J.C.: Harmonic maps from surfaces into projective spaces. Adv. Math. 49, 217–263 (1983)

    MathSciNet  MATH  Google Scholar 

  15. Fay, J.D.: Theta-Functions on Riemann Surfaces. Lecture Notes in Math. 352, Springer-Verlag, 1973

  16. Ferus, D., Pedit, F., Pinkall, U., Sterling, I.: Minimal tori in S4. J. Reine Angew. Math. 429, 1–47 (1992)

    MathSciNet  MATH  Google Scholar 

  17. Guest, M.A.: Harmonic Maps, Loop Groups, and Integral Systems. London Math. Soc. Student Texts 38, Cambridge University Press, Cambridge, 1997

  18. Haskins, M.: Special Lagrangian cones, math.DG/0005164, 2000

  19. Hashimoto, H., Taniguchi, T., Udagawa, S.: Constructions of almost complex 2-tori of type (III) in the nearly Kähler 6-shpere. Preprint, 2002

  20. Hitchin, N.: Harmonic maps from a 2-torus to the 3-sphere. J. Diff. Geom. 31, 627–710 (1990)

    MathSciNet  MATH  Google Scholar 

  21. Hitchin, N.: Integrable systems in Riemannian geometry. J. Diff. Geom. suppl. 4, 21–81 (1998)

    Google Scholar 

  22. Its, A.R.: Liouville’s theorem and the method of the inverse problem. J. Soviet Math. 31(6), 3330–3338 (1985)

    MATH  Google Scholar 

  23. Joyce, D.: Special Lagrangian 3-folds and integrable systems. math.DG/010249, 2001

  24. Li, A., Wang, C.: Geometry of surfaces in Preprint, 1999

  25. McIntosh, I.: Special Lagrangian cones in and primitive harmonic maps. math.DG/02011557, 2002

  26. Mikhailov, A.V.: The reduction problem and the inverse scattering method. Physica 3D, 73–117 (1981)

    Article  Google Scholar 

  27. Moerbeke, Van, P., Mumford, D.: The spectrum of difference operators and algebraic curves. Acta Math. 143, 93–154 (1979)

    MATH  Google Scholar 

  28. Natanzon, S.M.: Moduli of real algebraic surfaces, and their superanalogues. Differentials, spinors, and Jacobians of real curves. Russian Math. Surv. 54(6), 1091–1147 (1999)

    Google Scholar 

  29. Ohnita, Y.: Toda equations and harmonic maps. In: State of the art and perspectives in studies on nonlinear integrable systems (Japanese) (Kyoto, 1993). RIMS Kokyuroku, 868, 1994, pp. 66–73

  30. Pinkall, U., Sterling, I.: On the classification of constant mean curvature tori. Ann. Math. 130, 407–451 (1989)

    MathSciNet  MATH  Google Scholar 

  31. Sharipov, R.A.: Minimal tori in the five-dimensional sphere in . Theor. Math. Phys. 87(1), 363–369 (1991)

    MATH  Google Scholar 

  32. Tzitzéica, M.: Sur une nouvelle classe de surfaces. C. R. Acad. Sci. Paris 150, 955–956 (1910)

    Google Scholar 

  33. Wang, C.: The classification of homogeneous surfaces in Geometry and topology of submanifolds, X (Beijing/Berlin, 1999), World Sci. Publishing, River Edge, NJ, 2000, pp. 303–314

  34. Wente, H.: Counterexample to a conjecture of H. Hopf. Pacific J. Math. 121, 193–243 (1986)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Center for Geometry, Analysis, Numerics and Graphics, University of Massachusetts, Amherst, MA, 01003, USA

    Hui Ma

  2. KLMM, Institute of Systems Science, Chinese Academy of Sciences, Beijing, 100080, China

    Yujie Ma

Authors
  1. Hui Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yujie Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hui Ma.

Additional information

Mathematics Subject Classification (2000): 53C42, 53C43, 53D12

Supported by NSFC 10271004, Education Foundation of Tsinghua University and Sfb288 at TU Berlin.

Supported by the project No. G1998030601 of China.

Acknowledgement The authors are grateful to Professor Y. Ohnita for initiating their research in this field during his visit to Peking university in the autumn of 1998, Professors A.I. Bobenko, H.Z. Li, A.V. Mikhailov, F. Pedit, R.A. Sharipov, C.P. Wang and Y.J. Zhang for their assistance and Professors D. Joyce and I. McIntosh for their interests and helpful comments. The authors would like to express their sincere thanks to Professor W.H. Chen for his constant encouragements and guidance. They also thank the referee for several helpful suggestions and comments. Most of the results of this paper was reported by the first author during the second international conference on harmonic morphisms and harmonic maps at the CIRM in 2001. She would like to thank the organizers for their hospitality and the participants for their interests.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ma, H., Ma, Y. Totally real minimal tori in . Math. Z. 249, 241–267 (2005). https://doi.org/10.1007/s00209-004-0693-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-004-0693-5

Search

Navigation