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 , · · · , ϕα 0 : S 2 → G k,n be a linearly full nondegenerate pseudo-holomorphic harmonic sequence, and let degϕα and Kα be the degree and the Gauss curvature of ϕα (α = 0, 1, · · · , α 0 ) respectively. Assume that ϕ 0 , ϕ 1 , · · · , ϕα 0 is totally unramified. Then we prove that (i) degϕα = k(α 0 − 2α) for all α = 0, 1, · · · , α 0 ; (ii) Kα = 4 k(α 0 +2α(α 0 −α)) if Kα 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.

On conformal minimal 2-spheres in complex Grassmann manifold G(2,n)

“…Some of the functions in log(·) probably have isolated zeros, however, we assume them to have no zeros in the proof. The result (3) was proved by Jiao in his recent paper [11], and also by Zheng in [16] using different methods.…”

“…They classified the minimal two-spheres immersed in CP n and proved the rigidity theorems of conformal minimal two-spheres in CP n , but some of these properties are not inherited when the ambient space is G(k, n), k ≥ 2. The pseudoholomorphic two-spheres in G(k, n) were studied by Jiao and Peng [10], Jiao [11] and Zheng [16]. They got various pinching theorems about the Gaussian curvature and Kähler angle.…”

On holomorphic curves in a complex Grassmann manifold G(2, n)

“…Recently, we studied geometric properties of pseudo-holomorphic two-spheres in G(k, n)(cf. [5][6][7]). Let s : S 2 → G(k, n) be a smooth map.…”

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

“…where P = ∂ Az λ 2 with A z = (2φ − I)∂φ, I is the identity matrix (cf. [12,13]). In the following, we review the rigidity theorem of conformal minimal immersions with constant curvature from S 2 to CP N .…”