Conformal minimal immersions with constant curvature from S² to Q₅

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.

Keywords

Conformal minimal surface / isotropy order / constant curvature / linearly full

References

[1] Bahy-El-Dien A, Wood J C. The explicit construction of all harmonic two-spheres in G₂(ℝⁿ): J Reine Angew Math, 1989, 398: 36–66

[2] Bolton J, Jensen G R, Rigoli M, Woodward L M. On conformal minimal immersions of S² into ℂPⁿ: Math Ann, 1988, 279(4): 599–620

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

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

[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

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

[7] Jiao X X,Li M Y.Classification of conformal minimal immersions of constant curvature from S² to Q_n: Ann Mat Pura Appl, 2017, 196(3): 1001–1023

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

[9] Li M Y, Jiao X X, He L. Classification of conformal minimal immersions of constant curvature from S² to Q₃: J Math Soc Japan, 2016, 68(2): 863–883

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

[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

[12] Wang J, Jiao X X. Conformal minimal two-spheres in Q₂: Front Math China, 2011, 6(3): 535–544

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

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