Skip to main content
SpringerLink
Log in

Conformally flat, minimal, Lagrangian submanifolds in complex space forms

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

We investigate n-dimensional (n ⩾ 4), conformally flat, minimal, Lagrangian submanifolds of the n-dimensional complex space form in terms of the multiplicities of the eigenvalues of the Schouten tensor and classify those that admit at most one eigenvalue of multiplicity one. In the case where the ambient space is ℂn, the quasi umbilical case was studied in Blair (2007). However, the classification there is not complete and several examples are missing. Here, we complete (and extend) the classification and we also deal with the case where the ambient complex space form has non-vanishing holomorphic sectional curvature.

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. Blair D E. On Lagrangian catenoids. Canad Math Bull, 2007, 50: 321–333

    Article  MathSciNet  Google Scholar 

  2. Brozos-Vázquez M, Garcia-Rio E, Vázquez-Lorenzo R. Some remarks on locally conformally flat static space-times. J Math Phys, 2005, 46: 022501

    Article  MathSciNet  Google Scholar 

  3. Castro I, Urbano F. On a minimal Lagrangian submanifold of ℂn foliated by spheres. Michigan Math J, 1999, 46: 71–82

    Article  MathSciNet  Google Scholar 

  4. Chen B-Y. Riemannian geometry of Lagrangian submanifolds. Taiwanese J Math, 2001, 5: 681–723

    MathSciNet  MATH  Google Scholar 

  5. Cheng X, Hu Z, Moruz M, et al. On product minimal Lagrangian submanifolds in complex space forms. J Geom Anal, 2021, 31: 1934–1964

    Article  MathSciNet  Google Scholar 

  6. Derdziński A. Some remarks on the local structure of Codazzi tensors. In: Global Differential Geometry and Global Analysis. Lecture Notes in Mathematics, vol. 838. Berlin: Springer, 1981, 251–255

    Chapter  Google Scholar 

  7. Ejiri N. Totally real minimal immersions of n-dimensional real space forms into n-dimensional complex space forms. Proc Amer Math Soc, 1982, 84: 243–246

    MathSciNet  MATH  Google Scholar 

  8. Hiepko S. Eine innere Kennzeichnung der verzerrten Produkte. Math Ann, 1979, 241: 209–215

    Article  MathSciNet  Google Scholar 

  9. Lafontaine J. Conformal geometry from the Riemannian viewpoint. In: Conformal Geometry. Aspects of Mathematics, vol. 12. Wiesbaden: Vieweg+Teubner Verlag, 1988, 65–92

    Chapter  Google Scholar 

  10. Li H, Wang X. Calabi product Lagrangian immersions in complex projective space and complex hyperbolic space. Results Math, 2011, 59: 453–470

    Article  MathSciNet  Google Scholar 

  11. Reckziegel H. Horizontal lifts of isometric immersions into the bundle space of a pseudo-Riemannian submersion. In: Global Differential Geometry and Global Analysis. Lecture Notes in Mathematics, vol. 1156. Berlin: Springer, 1985, 264–279

    Google Scholar 

Download references

Acknowledgements

The first author was supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia (Grant No. 174012).

Author information

Authors and Affiliations

  1. Faculty of Mathematics, University of Belgrade, Belgrade, 11000, Serbia

    Miroslava Antić

  2. Laboratoire de Mathématiques pour l’Ingénieur, Université Polytechnique Hauts-de-France, Valenciennes, 59313, France

    Luc Vrancken

  3. Departement Wiskunde, Katholieke Universiteit Leuven, Leuven, 3001, Belgium

    Luc Vrancken

Authors
  1. Miroslava Antić
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luc Vrancken
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Miroslava Antić.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Antić, M., Vrancken, L. Conformally flat, minimal, Lagrangian submanifolds in complex space forms. Sci. China Math. 65, 1641–1660 (2022). https://doi.org/10.1007/s11425-021-1897-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-021-1897-0

Keywords

MSC(2020)

Search

Navigation