Conformal minimal two-spheres in Qn

In this paper we study, using moving frames, conformal minimal two-spheres S2 immersed into a complex hyperquadric Qn equipped with the induced Fubini-Study metric from a complex projective n+1-space CPn+1. Two associated functions τX and τY are introduced to classify the problem into several cases. It is proved that τX or τY must be identically zero if f: S2 → Qn is a conformal minimal immersion. Both the Gaussian curvature K and the Kähler angle θ are constant if the conformal immersion is totally geodesic. It is also shown that the conformal minimal immersion is totally geodesic holomorphic or antiholomorphic if K = 4. Excluding the case K = 4, conformal minimal immersion f: S2 → Q2 with Gaussian curvature K ⩾ 2 must be totally geodesic with (K, θ) ∈ {(2, 0), (2, 2/π), (2, π)}.