On Conformal Minimal Immersions of Two-Spheres in a Complex Hyperquadric with Parallel Second Fundamental Form

Abstract

In this paper we determine all conformal minimal immersions of twospheres in complex hyperquadric Q n with parallel second fundamental form.

Classification of conformal minimal immersions from $S^2$ to $G(2,N;\mathbb{C})$ with parallel second fundamental form

“…It is easy to check that these eighteen minimal immersions are all homogeneous. Of course they contain those given by ( [11], Theorem 1.1 and [14], Theorem 1.1), even more than those (cf. cases (1)(2) and (3) shown in Theorem 1.1 etc.…”

“…Recently, we discussed the geometry of conformal minimal immersions from S 2 to the hyperquadric Q n and gave a complete classification theorem of them under the assumption that they have parallel second fundamental form (cf. [14]). L. He and the first author also classified all conformal minimal two-spheres immersed in the quaternionic projective space HP n with parallel second fundamental form (cf.…”

“…It shows that φ : S 2 → G(2, N ; C) is totally real with ∇B = 0 and r = 1. Then from [1] and Theorem 1.1 of [14],…”

“…) is totally real with parallel second fundamental form and isotropy order r ≥ 2. Then from [1,11] and [14] , φ belongs to the following two cases:…”

Rigidity of Conformal Minimal Immersions of Constant Curvature from $$S^2$$ to $$Q_4$$

“…where P = ∂ Az [11]). In the following, we give a definition of the unramified harmonic map as follows.…”