Annals of Global Analysis and Geometry

, Volume 43, Issue 4, pp 299–329 | Cite as

Darboux transforms and spectral curves of constant mean curvature surfaces revisited

  • E. CarberryEmail author
  • K. Leschke
  • F. Pedit


We study the geometric properties of Darboux transforms of constant mean curvature (CMC) surfaces and use these transforms to obtain an algebro-geometric representation of constant mean curvature tori. We find that the space of all Darboux transforms of a CMC torus has a natural subset which is an algebraic curve (called the spectral curve) and that all Darboux transforms represented by points on the spectral curve are themselves CMC tori. The spectral curve obtained using Darboux transforms is not bi-rational to, but has the same normalisation as, the spectral curve obtained using a more traditional integrable systems approach.


Constant mean curvature surfaces Harmonic maps Integrable systems 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bernstein H.: Non-special, non-canal isothermic tori with spherical lines of curvature. Trans. Amer. Math. Soc. 353, 2245–2274 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bobenko A.I.: All constant mean curvature tori in R 3, S 3 and H 3 in terms of theta-functions. Math. Ann. 290(2), 209–245 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bohle, C.: Constrained Willmore tori in the 4-sphere. arXiv:0803.0633v1Google Scholar
  4. 4.
    Bohle, C., Leschke, K., Pedit, F., Pinkall, U.: (2007) Conformal maps from a 2-torus to the 4-sphere. J. Reine Angew. Math. doi: 10.1515/crelle.2011.156
  5. 5.
    Bohle C., Pedit F., Pinkall U.: The spectral curve of a quaternionic holomorphic line bundle over a 2-torus. Manuscripta Math. 130(3), 311–352 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Burstall, F., Ferus, D., Leschke, K., Pedit, F., Pinkall, U.: Conformal geometry of surfaces in S 4 and quaternions, Lecture Notes in Mathematics, Springer, Berlin (2002)Google Scholar
  7. 7.
    Darboux G.: Sur les surfaces isothermiques. C. R. Acad. Sci. Paris 128, 1299–1305 (1899)zbMATHGoogle Scholar
  8. 8.
    Ercolani, N.M., Knörrer, H., Trubowitz, E.: Hyperelliptic curves that generate constant mean curvature tori in \({\mathbb{R}^3}\) . Integrable systems (Luminy 1991), Progr. Math. 115 81–114 (1993)Google Scholar
  9. 9.
    Ferus D., Leschke K., Pedit F., Pinkall U.: Quaternionic holomorphic geometry: Plücker formula, Dirac eigenvalue estimates and energy estimates of harmonic 2-tori. Invent. Math. 146, 507–593 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Große-Brauckmann, K., Kusner, R.B., Sullivan, J.M.: Triunduloids: embedded constant mean curvature surfaces with three ends and genus zero. J. Reine Angew. Math. 564, 35–61 (2003). arXiv:math/0102183Google Scholar
  11. 11.
    Hertrich-Jeromin, U.: Introduction to Möbius differential geometry. London Mathematical Society Lecture Note Series 300. Cambridge University Press, Cambridge (2003)Google Scholar
  12. 12.
    Hertrich-Jeromin U., Pedit F.: Remarks on the Darboux transform of isothermic surfaces. Doc. Math. 2, 313–333 (1997)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Hitchin N.: Harmonic maps from a 2-torus to the 3-sphere. J. Differential Geom. 31, 627–710 (1990)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Jaggy C.: On the classification of constant mean curvature tori in R 3. Comment. Math. Helv. 69(4), 640–658 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Kapouleas N.: Compact constant mean curvature surfaces in euclidean three-space. J. Differential Geom. 33(3), 683–715 (1991)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Leschke, K.: Transformations on Willmore surfaces, Habilitationsschrift, Universität Augsburg, Augsburg (2006)Google Scholar
  17. 17.
    Leschke, K.: Families of conformal tori of revolution in the 3-sphere. In: Ohnita, Y. (ed.) Proceedings of the 16th OCU International Academic Symposium 2008. Riemann Surfaces, Harmonic Maps and Visualization. OCAMI Studies, vol. 3, pp. 91–103. (2009) arXiv:0909.3965Google Scholar
  18. 18.
    Leschke, K., Romon, P.: Spectral curve of Hamiltonian stationary tori. Calc. Var. Partial Differential Equations 38(1–2), 45–74 (2010). arXiv:0806.1848Google Scholar
  19. 19.
    Mazzeo, R., Pacard, F.: Constant mean curvature surfaces with Delaunay ends. Comm. Anal. Geom. 9(1), 169–237 (2001). arXiv:math/9807039Google Scholar
  20. 20.
    Pinkall U., Sterling I.: On the classification of constant mean curvature tori. Ann. of Math. 130(2), 407–451 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Pohlmeyer K.: Integrable Hamiltonian systems and interactions through quadratic constraints. Comm. Math. Phys. 46, 207–221 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Ruh E., Vilms J.: The tension field of the Gauss map. Trans. Amer. Math. Soc. 149, 569–573 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Schmidt, M.: A proof of the Willmore conjecture (2002). arxiv:0203224Google Scholar
  24. 24.
    Sterling I., Wente H.: Existence and classification of constant mean curvature multibubbletons of finite and infinite type. Indiana Univ. Math. J. 42, 1239–1266 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Taimanov, I.: The Weierstrass representation of spheres in \({\mathbb{R}^3}\) , the Willmore numbers, and soliton spheres. Proc. Steklov Inst. Math 225, 322–343 (1999). arXiv:math/9801022Google Scholar
  26. 26.
    Uhlenbeck K.: Harmonic maps into Lie groups: classical solutions of the chiral model. J. Differential Geom. 30(1), 1–50 (1989)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Wente Henry C.: Counterexample to a conjecture of H. Hopf. Pacific J. Math. 121(1), 193–243 (1986)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of SydneySydneyAustralia
  2. 2.Department of MathematicsUniversity of LeicesterLeicesterUK
  3. 3.Mathematisches InstitutTübingenGermany
  4. 4.Department of Mathematics and StatisticsUniversity of Massachusetts AmherstAmherstUSA

Personalised recommendations