Conformal minimal two-spheres in Q _n

Abstract

In this paper we study, using moving frames, conformal minimal two-spheres S ² immersed into a complex hyperquadric Q _n equipped with the induced Fubini-Study metric from a complex projective n+1-space CP ⁿ⁺¹. Two associated functions τ _X and τ _Y are introduced to classify the problem into several cases. It is proved that τ _X or τ _Y must be identically zero if f: S ² → Q _n 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: S ² → Q ₂ with Gaussian curvature K ⩾ 2 must be totally geodesic with (K, θ) ∈ {(2, 0), (2, 2/π), (2, π)}.