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

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