Skip to main content
SpringerLink
Log in

Rigidity of Homogeneous Holomorphic \(S^2\) in a Complex Grassmann Manifold G(2, N)

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

In this paper, we give a local rigid characterization of all homogeneous holomorphic two-spheres in a complex Grassmann manifold G(2, N). Let \(\kappa \) denote the new global invariant defined in terms of the square norm of (1, 0) part of the second-order covariant differential of the first \(\partial \)-transform for a holomorphic curve in G(2, N). It is proved that a linearly full non-degenerate holomorphic curve of constant curvature and constant square norm of the second fundamental form in G(2, N) with \(\kappa =0\) must be homogeneous.

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. Bando, S., Ohnita, Y.: Minimal \(2\)-spheres with constant curvature in \({\mathbb{C} }P^n\). J. Math. Soc. Jpn. 3, 477–487 (1987)

    MATH  Google Scholar 

  2. Bolton, J., Jensen, G.R., Rigoli, M., Woodward, L.M.: On conformal minimal immersions of \(S^2\) into \({\mathbb{C} }P^n\). Math. Ann. 279, 599–620 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  3. Burstall, F.E., Wood, J.C.: The construction of harmonic maps into complex Grassmannians. J. Differ. Geom. 23, 255–297 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  4. Calabi, E.: Isometric embedding of complex manifolds. Ann. Math. 58, 1–23 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chern, S.S., M. do Carmo, S. Kobayashi: Minimal submanifolds of a sphere with second fundamental form of constant length. In: Functional Analysis and Related Fields (Proceeding of Conference in Honor of M. Stone at University of Chicago), vol. 59–75, p. 1970. Springer, Berlin (1968)

  6. Chern, S.S., Wolfson, J.G.: Harmonic maps of the two-sphere into a complex Grassmann manifold II. Ann. Math. 125, 301–335 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chi, Q.S., Xie, Z.X., Xu, Y.: Fano 3-folds and classification of constantly curved holomorphic 2-spheres of degree 6 in the complex Grassmannian\(G(2,5)\). arXiv:2208.08525

  8. Chi, Q.S., Zheng, Y.B.: Rigidity of pseudo-holomorphic curves of constant curvature in Grassmann manifold. Trans. Am. Math. Soc. 313, 393–406 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  9. Din, A.M., Zakrzewski, W.J.: General classical solutions in the \({\mathbb{C} }P^n\) model. Nucl. Phys. B 174, 397–406 (1980)

    Article  Google Scholar 

  10. Delisle, L., Hussin, V., Zakrzewski, W.J.: Constant curvature solutions of Grassmannian sigma models: (1) holomorphic solutions. J. Geom. Phys. 66, 24–36 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Delisle, L., Hussin, V., Zakrzewski, W.J.: Constant curvature solutions of Grassmannian sigma models: (2) non-holomorphic solutions. J. Geom. Phys. 71, 1–10 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Fei, J., Jiao, X.X., Xu, X.W.: On conformal minimal 2-spheres in complex Grassmann manifold \(G(2,n)\). Proc. Indian Acad. Sci. (Math. Sci.) 121, 181–199 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fei, J., Jiao, X.X., Xiao, L., Xu, X.W.: On the classification of homogeneous \(2\)-spheres in the complex Grassmannians. Osaka J. Math. 50, 135–152 (2013)

    MathSciNet  MATH  Google Scholar 

  15. Fei, J., He, L.: On holomorphic two-spheres with constant curvature in the complex Grassmann manifold \(G(2, n)\). arXiv:1903.09990

  16. Fei, J., Xu, X.W.: Local rigidity of holomorphic curves in the complex Grassmann manifold \(G(2,6)\). J. Geom. Phys. 121, 438–451 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fei, J.: Classification of homogeneous holomorphic two-spheres in complex Grassmann manifolds. Differ. Geom. Appl. 62, 1–38 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  18. 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. 41, 775–814 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  19. He, L.: The conformal minimal immersions of \(S^2\)in\(G(k,n)\), Ph.D Thesis. University of Chinese Academy Sciences (2014)

  20. He, L.: Degree of constantly curved holomorphic 2-spheres in the complex Grassmannians \(G(2,n+2;{\mathbb{C}})\). arXiv:2204.08116

  21. He, L., Jiao, X.X., Zhou, X.C.: Rigidity of holomorphic curves of constant curvature in \(G(2,5)\). Differ. Geom. Appl. 43, 21–44 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jiao, X.X.: Pseudo-holomorphic curves of constant curvature in complex Grassmannians. Isr. J. Math. 163, 45–60 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Jiao, X.X., Peng, J.G.: Classification of holomorphic two-spheres with constant curvature in complex Grassmann manifold \(G(2,5)\). Differ. Geom. Appl. 20, 267–277 (2004)

    Article  Google Scholar 

  24. Jiao, X.X., Peng, J.G.: On holomorphic curves of constant curvature in the complex Grassmann manifold \(G(2,5)\). Acta Math. Sin. 31B(1), 237–248 (2011)

    MathSciNet  MATH  Google Scholar 

  25. Li, Z.Q., Jin, M.M.: Constant curved holomorphic \(2\)-spheres in \(G(2,4)\). J. Math. Sci. Adv. Appl. 1, 547–561 (2008)

    MathSciNet  MATH  Google Scholar 

  26. Li, Z.Q., Yu, Z.H.: Constant curved minimal \(2\)-spheres in \(G(2,4)\). Manuscr. Math. 100, 305–316 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  27. Peng, C., Xu, X.W.: Classification of minimal homogeneous two spheres in the complex Grassmann manifold \(G(2, n)\). J. Math. Pures Appl. 103(2), 374–399 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Uhlenbeck, K.: Harmonic maps into Lie groups (classical solution of the chiral model). J. Differ. Geom. 30, 1–50 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wang, J., Fei, J.: Rigidity theorems for holomorphic curves in a complex Grassmann manifold \(G(3, 6)\). Int. J. Math. 32, 2150095 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  30. Wood, J.C.: The explicit construction and parametrization of all harmonic maps from two-sphere to a complex Grassmannian. J. Reine Angew. Math. 386, 1–31 (1988)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to express gratitude for the referee’s valuable comments and suggestions. This work was supported by National Key R &D Program of China (Grant No. 2022YFA1006600) and NSFC (Grant Nos. 11401481,  12071338,  12071352, 11301273, 11971237). The first named author was also supported by the Research Enhancement Fund of Xi’an Jiaotong-Liverpool University (REF-18-01-03). The third named author was also supported by the Natural Science Foundation of Jiangsu Province (Grant No. BK20221320).

Author information

Authors and Affiliations

  1. School of Mathematics and Physics, Xi’an Jiaotong-Liverpool University, Suzhou, 215123, People’s Republic of China

    Jie Fei

  2. Center for Applied Mathematics, Tianjin University, Tianjin, 300072, People’s Republic of China

    Ling He

  3. School of Mathematics Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing, 210023, People’s Republic of China

    Jun Wang

Authors
  1. Jie Fei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ling He
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jun Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jun Wang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fei, J., He, L. & Wang, J. Rigidity of Homogeneous Holomorphic \(S^2\) in a Complex Grassmann Manifold G(2, N). J Geom Anal 33, 324 (2023). https://doi.org/10.1007/s12220-023-01387-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12220-023-01387-7

Keywords

Mathematics Subject Classification

Search

Navigation