Frontiers of Mathematics in China

, Volume 14, Issue 2, pp 315–348 | Cite as

Conformal minimal immersions with constant curvature from S2 to Q5

  • Xiaoxiang Jiao
  • Hong LiEmail author
Research Article


We study the geometry of conformal minimal two spheres immersed in G(2; 7; ℝ): Then we classify the linearly full irreducible conformal minimal immersions with constant curvature from S2 to G(2; 7; ℝ); or equivalently, a complex hyperquadric Q5 under some conditions. We also completely determine the Gaussian curvature of all linearly full totally unramified irreducible and all linearly full reducible conformal minimal immersions from S2 to G(2; 7; ℝ) with constant curvature. For reducible case, we give some examples, up to SO(7) equivalence, in which none of the spheres are congruent, with the same Gaussian curvature.


Conformal minimal surface isotropy order constant curvature linearly full 


53C42 53C55 


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This work was supported by the National Natural Science Foundation of China (Grant No. 11871450)


  1. 1.
    Bahy-El-Dien A, Wood J C. The explicit construction of all harmonic two-spheres in G 2(ℝn). J Reine Angew Math, 1989, 398: 36–66MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bolton J, Jensen G R, Rigoli M, Woodward L M. On conformal minimal immersions of S 2 into ℂPn. Math Ann, 1988, 279(4): 599–620MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Burstall F E, Wood J C. The construction of harmonic maps into complex Grassmannians. J Differential Geom, 1986, 23(3): 255–297MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Eells J, Sampson J H. Harmonic mappings of Riemannian manifolds. Amer J Math, 1964, 86(1): 109–160MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Erdem S, Wood J C. On the construction of harmonic maps into a Grassmannian. J Lond Math Soc (2), 1983, 28(1): 161–174MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Jiao X X. Pseudo-holomorphic curves of constant curvature in complex Grassmannians. Israel J Math, 2008, 163(1): 45–60MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Jiao X X, Li M Y. Classification of conformal minimal immersions of constant curvature from S 2 to Q n. Ann Mat Pura Appl, 2017, 196(3): 1001–1023MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Jiao X X, Peng J G. Minimal two-sphere in G(2; 4). Front Math China, 2010, 5(2): 297–310MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Li M Y, Jiao X X, He L. Classification of conformal minimal immersions of constant curvature from S 2 to Q 3. J Math Soc Japan, 2016, 68(2): 863–883MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Li Z Q, Yu Z H. Constant curved minimal 2-spheres in G(2; 4). Manuscripta Math, 1999, 100(3): 305–316MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Peng C K, Wang J, Xu X W. Minimal two-spheres with constant curvature in the complex hyperquadric. J Math Pures Appl, 2016, 106(3): 453–476MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Wang J, Jiao X X. Conformal minimal two-spheres in Q 2. Front Math China, 2011, 6(3): 535–544MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Wolfson J G. Harmonic maps of the two-sphere into the complex hyperquadric. J Differential Geom, 1986, 24(2): 141–152MathSciNetCrossRefzbMATHGoogle Scholar

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© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina

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