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 degφα and K α be the degree and the Gauss curvature of φα (α = 0, 1, ⋯, α 0) respectively. Assume that φ 0, φ 1, ⋯, $$\varphi _{\alpha _0 } $$ is totally unramified. Then we prove that (i) degφα 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 Kahler angle in complex Grassmiannians to be of constant curvature.