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

Abstract

For a harmonic map f from a Riemann surface into a complex Grassmann manifold, Chern and Wolfson [4] constructed new harmonic maps ∂ f and ∂ f through the fundamental collineations ∂ and∂ respectively. In this paper, we study the linearly full conformal minimal immersions from S 2 into complex Grassmannians G(2, n), according to the relationships between the images of ∂ f and∂ f . We obtain various pinching theorems and existence theorems about the Gaussian curvature, Kähler angle associated to the given minimal immersions, and characterize some immersions under special conditions. Some examples are given to show that the hypotheses in our theorems are reasonable.

Constant curvature solutions of Grassmannian sigma models: (1) Holomorphic solutions

“…As a final comment in this subsection let us mention that we know from Theorem 1 that the complete classification of such solutions for G (2,5) gives rise to a complete classification of holomorphic solutions with constant curvatures for G(3, 5).…”

“…It remains to prove Conjecture 1 which can be reformulated, in this special case, as the following proposition: Proof: In the expression of detM (2,5) given as (80), we can see that the highest powers of |x + | are r = r 3 + r 2 + s 1 − 1 and r 3 + s 1 with the corresponding coefficients γ 23 and γ 13 in the Plücker coordinates.…”

“…Since the appearance of this paper many other papers have also been written. In particular, a classification of holomorphic spheres with constant curvature has been produced 2 for G (2,5). Other contributions by the same authors mostly dealt with non-holomorphic immersions 3,4,5 for G (2, n).…”

On holomorphic two-spheres with constant curvature in the complex Grassmann manifold G(2,n)

“…In this subsection, we introduce the geometry of holomorphic curves in G(2, n) by the method of moving frames. More details can be found in [7] and [12]. Let ϕ be a linearly full holomorphic immersion from S 2 into G(2, n).…”

“…invariants of analytic type on S 2 vanishing only at isolated points, and away from their zeros, they satisfy (cf. [7], [12])…”