Annals of Global Analysis and Geometry

, Volume 53, Issue 4, pp 583–595 | Cite as

An energy functional for Lagrangian tori in \(\mathbb {C}P^2\)

  • Hui Ma
  • Andrey E. Mironov
  • Dafeng Zuo


A two-dimensional periodic Schrödingier operator is associated with every Lagrangian torus in the complex projective plane \({\mathbb C}P^2\). Using this operator, we introduce an energy functional on the set of Lagrangian tori. It turns out this energy functional coincides with the Willmore functional \(W^{-}\) introduced by Montiel and Urbano. We study the energy functional on a family of Hamiltonian-minimal Lagrangian tori and support the Montiel–Urbano conjecture that the minimum of the functional is achieved by the Clifford torus. We also study deformations of minimal Lagrangian tori and show that if a deformation preserves the conformal type of the torus, then it also preserves the area, i.e., preserves the value of the energy functional. In particular, the deformations generated by Novikov–Veselov equations preserve the area of minimal Lagrangian tori.


Lagrangian surfaces Energy functional Novikov–Veselov hierarchy 



The authors are grateful to professor Yong Luo and professor Iskander Taimanov for their useful discussions.


  1. 1.
    Carberry, E., McIntosh, I.: Minimal Lagrangian 2-tori in \({\mathbb{C}}P^2\) come in real families of every dimension. J. Lond. Math. Soc. (2) 69(2), 531–544 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Castro, I., Urbano, F.: New examples of minimal Lagrangian tori in the complex projective plane. Manuscr. Math. 85, 265–281 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dorfmeister, J.F., Ma, H.: Explicit expressions for the Iwasawafactors, the metric and the monodromy matrices for minimal Lagrangian surfaces in \(\mathbb{C}P^2\). In: Dynamical Systems, Numbertheory and Applications, pp. 19–47. World Science Publishing, Hackensack (2016)Google Scholar
  4. 4.
    Dorfmeister, J.F., Ma, H.: A new look at equivariant minimal Lagrangian surfaces in \(\mathbb{C}P^2\). In: Geometry and Topology of Manifolds, pp. 97–125. Springer Proc. Math. Stat., vol. 154. Springer, Berlin (2016)Google Scholar
  5. 5.
    Haskins, H.: The geometric complexity of special Lagrangian \(T^2\)-cones. Invent. Math. 157(1), 11–70 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Joyce, D.: Special Lagrangian submanifolds with isolated conical singularities. v. Survey and applications. J. Differ. Geom. 63(2), 279–347 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ma, H., Ma, Y.: Totally real minimal tori in \(\mathbb{C}P^2\). Math. Z. 249(2), 241–267 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ma, H.: Hamiltonian stationary Lagrangian surfaces in \(\mathbb{C}P^2\). Ann. Glob. Anal. Geom. 27, 1–16 (2005)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ma, H., Schmies, M.: Examples of Hamiltonian stationary Lagrangian tori in \(\mathbb{C}P^2\). Geom. Dedicata 118, 173–183 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Marques, F.C., Neves, A.: Min–max theory and the Willmore conjecture. Ann. Math. 2(179), 683–782 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Mironov, A.E.: The Novikov–Veselov hierarchy of equations and integrable deformations of minimal Lagrangian tori in \(\mathbb{C}P^2\). (Russian) Sib. Elektron. Mat. Izv. 1, 38–46 (also arXiv:math/0607700v1) (2004)
  12. 12.
    Mironov, A.E.: New examples of Hamilton-minimal and minimal Lagrangian submanifolds in \({\mathbb{C}}^n\) and \({\mathbb{C}}P^n.\). Sb. Math. 195(1), 85–96 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Mironov, A.E.: Relationship between symmetries of the Tzitzéica equation and the Novikov–Veselov hierarchy. Math. Notes 82(4), 569–572 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Montiel, S., Urbano, F.: A Willmore functional for compact surfaces in the complex projective plane. J. Reine Angew. Math. 546, 139–154 (2002)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Oh, Y.: Volume minimization of Lagrangian submanifolds under Hamiltonian deformations. Math. Z. 212, 175–192 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Sharipov, R.A.: Minimal tori in the five-dimensional sphere in \({\mathbb{C}}^3\). Theor. Math. Phys. 87(1), 363–369 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Taimanov, I.A.: Two-dimensional Dirac operator and the theory of surfaces. Rus. Math. Surv. 61(1), 79–159 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Taimanov, I.A.: Modified Novikov–Veselov equation and differential geometry of surfaces. Am. Math. Soc. Transl. Ser. 2 179, 133–151 (1997)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Veselov, A.P., Novikov, S.P.: Finite-gap two-dimensional potential Schrödinger operators. Explicit formulas and evolution equations (Russian). Dokl. Akad. Nauk SSSR 279(1), 20–24 (1984)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Mathematical SciencesTsinghua UniversityBeijingPeople’s Republic of China
  2. 2.Sobolev Institute of MathematicsNovosibirskRussia
  3. 3.Novosibirsk State UniversityNovosibirskRussia
  4. 4.School of Mathematical ScienceUniversity of Science and Technology of ChinaHefeiPeople’s Republic of China
  5. 5.Wu Wen-Tsun Key Laboratory of Mathematics, USTCChinese Academy of SciencesHefeiPeople’s Republic of China

Personalised recommendations