Minimal two-spheres in G(2, 4)

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: S² → ℂP³ (called its directrix curve) and then, it is proved that such a conformal minimal immersion ϕ: S² → G(2, 4) with constant curvature has constant Käahler angle. Furthermore, ϕ is either V₁ ⁽³⁾ + V₃ ⁽³⁾, which is totally geodesic, with constant Gauss curvature 2/5 and constant Käahler angle given by t = 3/2 or V₃ ⁽³⁾ + V₂ ⁽³⁾, which is totally real, but it is not totally geodesic, with constant Gauss curvature 2/3, where V₀ ⁽³⁾, V₁ ⁽³⁾, V₂ ⁽³⁾, V₃ ⁽³⁾: S² → ℂP³ is the Veronese sequence.

Keywords

Harmonic map / conformal minimal immersion / Gauss curvature / Kähler angle / complex Grassmann manifold

References

[1] Bolton J., Jensen G. R., Rigoli M., Woodward L. M. On conformal minimal immersions of S² into ℂP^N. Math Ann, 1988, 2794: 599-620.

[2] Chern S. S., Wolfson J. G. Minimal surfaces by moving frames. Amer J Math, 1983, 105: 59-83.

[3] Chern S. S., Wolfson J. G. Harmonic maps of the two-sphere in a complex Grassmann manifold. Ann of Math, 1987, 125: 301-335.

[4] 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.

[5] Eells J., Wood J. C. Harmonic maps from surfaces to complex projective space. Adv Math, 1983, 49: 217-263.

[6] Jiao X. X., Peng J. G. Pseudo-holomorphic curves in complex Grassmann manifolds. Trans Amer Math Soc, 2003, 355: 3715-3726.

[7] Jiao X. X., Peng J. G. On some conformal minimal two-spheres in a complex projective space. Quart J Math, 2010, 61: 87-101.

[8] Kenmotsu K., Masuda K. On minimal surfaces of constant curvature in two-dimensional complex space form. J Reine Angew Math, 2000, 523: 69-101.

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

[10] Ramanathan J. Harmonic maps from S² to G_2,4. J Diff Geom, 1984, 19: 207-219.

[11] Uhlenbeck K. Harmonic maps into Lie groups (classical solutions of the chiral model). J Diff Geom, 1989, 30: 1-50.

[12] Wolfson J. G. Harmonic sequences and harmonic maps of surfaces into complex Grassmann manifolds. J Diff Geom, 1988, 27: 161-178.

[13] Zheng Y. B. Quantization of curvature of harmonic two-spheres in Grassmann manifolds. Trans Amer Math Soc, 1989, 316: 193-214.