Abstract
In this paper, we mainly study the geometry of conformal minimal immersions of two-spheres in a complex Grassmann manifold G(2,4). At first, we give a precise description of any non-±holomorphic harmonic 2-sphere in G(2,4) with the linearly full holomorphic maps ψ0: S 2 → ℂP3 (called its directrix curve) and then, it is proved that such a conformal minimal immersion ϕ: S 2 → G(2, 4) with constant curvature has constant Käahler angle. Furthermore, ϕ is either V (3)1 + V (3)3 , which is totally geodesic, with constant Gauss curvature 2/5 and constant Käahler angle given by t = 3/2 or V (3)3 + V (3)2 , which is totally real, but it is not totally geodesic, with constant Gauss curvature 2/3, where V (3)0 , V (3)1 , V (3)2 , V (3)3 : S 2 → ℂP3 is the Veronese sequence.
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References
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
Chern S S, Wolfson J G. Minimal surfaces by moving frames. Amer J Math, 1983, 105: 59–83
Chern S S, Wolfson J G. Harmonic maps of the two-sphere in a complex Grassmann manifold. Ann of Math, 1987, 125: 301–335
Chi Q S, Zheng Y B. Rigidity of pseudo-holomorphic curves of constant curvature in Grassmann manifolds. Trans Amer Math Soc, 1989, 313: 393–406
Eells J, Wood J C. Harmonic maps from surfaces to complex projective space. Adv Math, 1983, 49: 217–263
Jiao X X, Peng J G. Pseudo-holomorphic curves in complex Grassmann manifolds. Trans Amer Math Soc, 2003, 355: 3715–3726
Jiao X X, Peng J G. On some conformal minimal two-spheres in a complex projective space. Quart J Math, 2010, 61: 87–101
Kenmotsu K, Masuda K. On minimal surfaces of constant curvature in two-dimensional complex space form. J Reine Angew Math, 2000, 523: 69–101
Li Z Q, Yu Z H. Constant curved minimal 2-spheres in G(2,4). Manuscripta Math, 1999, 100: 305–316
Ramanathan J. Harmonic maps from S 2 to G 2,4. J Diff Geom, 1984, 19: 207–219
Uhlenbeck K. Harmonic maps into Lie groups (classical solutions of the chiral model). J Diff Geom, 1989, 30: 1–50
Wolfson J G. Harmonic sequences and harmonic maps of surfaces into complex Grassmann manifolds. J Diff Geom, 1988, 27: 161–178
Zheng Y B. Quantization of curvature of harmonic two-spheres in Grassmann manifolds. Trans Amer Math Soc, 1989, 316: 193–214
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Jiao, X., Peng, J. Minimal two-spheres in G(2, 4). Front. Math. China 5, 297–310 (2010). https://doi.org/10.1007/s11464-010-0009-5
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DOI: https://doi.org/10.1007/s11464-010-0009-5