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Conformal minimal immersions with constant curvature from S2 to Q5

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Abstract

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.

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References

  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: 36–66

    MathSciNet  MATH  Google Scholar 

  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): 599–620

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Eells J, Sampson J H. Harmonic mappings of Riemannian manifolds. Amer J Math, 1964, 86(1): 109–160

    Article  MathSciNet  MATH  Google Scholar 

  5. Erdem S, Wood J C. On the construction of harmonic maps into a Grassmannian. J Lond Math Soc (2), 1983, 28(1): 161–174

    Article  MathSciNet  MATH  Google Scholar 

  6. Jiao X X. Pseudo-holomorphic curves of constant curvature in complex Grassmannians. Israel J Math, 2008, 163(1): 45–60

    Article  MathSciNet  MATH  Google Scholar 

  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): 1001–1023

    Article  MathSciNet  MATH  Google Scholar 

  8. Jiao X X, Peng J G. Minimal two-sphere in G(2; 4). Front Math China, 2010, 5(2): 297–310

    Article  MathSciNet  MATH  Google Scholar 

  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): 863–883

    Article  MathSciNet  MATH  Google Scholar 

  10. Li Z Q, Yu Z H. Constant curved minimal 2-spheres in G(2; 4). Manuscripta Math, 1999, 100(3): 305–316

    Article  MathSciNet  MATH  Google Scholar 

  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): 453–476

    Article  MathSciNet  MATH  Google Scholar 

  12. Wang J, Jiao X X. Conformal minimal two-spheres in Q 2. Front Math China, 2011, 6(3): 535–544

    Article  MathSciNet  MATH  Google Scholar 

  13. Wolfson J G. Harmonic maps of the two-sphere into the complex hyperquadric. J Differential Geom, 1986, 24(2): 141–152

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

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

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Correspondence to Hong Li.

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Jiao, X., Li, H. Conformal minimal immersions with constant curvature from S2 to Q5. Front. Math. China 14, 315–348 (2019). https://doi.org/10.1007/s11464-019-0763-y

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  • DOI: https://doi.org/10.1007/s11464-019-0763-y

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