Pseudo-holomorphic curves in complex Grassmann manifolds

Abstract

Abstract. It is proved that the Kähler angle of the pseudo-holomorphic sphere of constant curvature in complex Grassmannians is constant. At the same time we also prove several pinching theorems for the curvature and the Kähler angle of the pseudo-holomorphic spheres in complex Grassmannians with non-degenerate associated harmonic sequence.

Minimal two-spheres in G(2, 4)

“…Then, Zheng [13] showed that, for a minimal two-sphere in G(2,4), if its curvature K 2, then K is either 2 or 4, and all these maps are explicitly classified up to U(4)-congruences. In 2003, we proved that the Kähler angle of the pseudo-holomorphic curve of constant curvature in G(k, n) must be constant [6].…”

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.…”

Pseudo-holomorphic curves of constant curvature in complex Grassmannians

“…By [12] we get that the induced metric ds 2 1 by ϕ 1 is ds 2 0 + ds 2 2 . It can easily be checked that curvature K(ϕ 0 ) of ϕ 0 is 1, and ϕ 1 and ϕ 2 are not maps of constant curvature, namely, ϕ 0 , ϕ 1 , ϕ 2 : S 2 → G 2,6 is not a harmonic sequence of constant curvature.…”

“…In [8] and [12] pseudo-holomorphic curves in a complex Grassmann manifold were studied. Recently, the classification of holomorphic spheres of constant curvature in G 2,5 was investigated (cf., [11]).…”

“…. ., ϕ α0 : S 2 → G k,n (cf., [12]). Let degϕ α and K α be the degree and the curvature of ϕ α respectively.…”