Minimal two-spheres in G(2, 4)

Abstract

In this paper, we mainly study the geometry of conformal minimal immersions of two-spheres in a complex Grassmann manifold G(2,4). At first, we give a precise description of any non-±holomorphic harmonic 2-sphere in G(2,4) with the linearly full holomorphic maps ψ 0 : S 2 → CP 3 (called its directrix curve) and then, it is proved that such a conformal minimal immersion ϕ : S 2 → G(2, 4) with constant curvature has constant Kähler angle. Furthermore, ϕ is either V3 , which is totally geodesic, with constant Gauss curvature 2/5 and constant Kähler angle given by t = 3/2 or V2 , which is totally real, but it is not totally geodesic, with constant Gauss curvature 2/3, where V3 : S 2 → CP 3 is the Veronese sequence.

Geometry of surfaces associated to Grassmannian sigma models

“…In the holomorphic case [1], we presented some conjectures and constructed some solutions which are not related to the Veronese ones. These results clarified and extended results obtained elsewhere [3][4][5][6][7].…”

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

“…Let us now consider the case of r 3 + s 1 = 7. We get the following not equivalent choices for r 2 and s 1 : (r 2 , s 1 ) = (2, 2), (2, 3), (2,4) and (3,2). It is trivial to check that they are not compatible with the constraint det M = (1 + |x| 2 ) 7 .…”

“…Just over 10 years ago Li and Yu discussed, in a very interesting paper 1 , the classification of minimal 2-spheres with constant Gaussian curvatures (called 'curvature' in this paper) immersed in the complex Grassmannian manifold G (2,4).…”

“…Theorem A discussed the holomorphic case and it showed that only four constant curvatures were possible. These are given as K = 4, 2, 4 3 and 1. Moreover, their paper also gave explicit examples of such holomorphic immersions.…”

“…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).…”