Pseudo-holomorphic curves of constant curvature in complex Grassmannians

Abstract

In this paper we consider pseudo-holomorphic curves in complex Grassmiannians. Let φ ₀, φ ₁, ⋯, $\varphi _{\alpha _0 }$: S ² → G _k,n be a linearly full non-degenerate pseudo-holomorphic harmonic sequence, and let degφ_α and K _α be the degree and the Gauss curvature of φ_α (α = 0, 1, ⋯, α ₀) respectively. Assume that φ ₀, φ ₁, ⋯, $\varphi _{\alpha _0 }$ is totally unramified. Then we prove that (i) degφ_α for all α = 0, 1, ⋯, α ₀; (ii) $K_\alpha = \tfrac{4}{{k(\alpha _0 + 2\alpha (\alpha _0 - \alpha ))}}$ if K _α is constant for some α = 0, 1, ⋯, α ₀,. We also give some conditions for pseudo-holomorphic curves with constant Kähler angle in complex Grassmiannians to be of constant curvature.