Rigidity of holomorphic curves of constant curvature in G(2,5)

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

“…Since ϕ is unramified (cf. [18], Definition 2.4) with constant curvature K. Then we can choose the canonical coordinate on S 2 such that…”

“…In Theorem 1.1, if the condition of ϕ being unramified is improved to being totally unramified, that is, every element of the harmonic sequence generated by ϕ is unramified (cf. [18], Definition 2.4), then d = n − 3 and K = 4 n−2 . This result was firstly obtained by Jiao and Yu in [23].…”

“…And for non-trivial solutions of h, we can check that the corresponding holomorphic curve are not homogeneous. In fact, in [18] the second author and her co-authors have given the symmetric solutions of the polynomial equation (1.1) in the case n = 5, then determine all totally unramified holomorphic two-spheres of constant curvature in G(2, 5), which include two families of non-homogeneous holomorphic two-spheres.…”

“…But the solutions of (1.1) do not guarantee that the corresponding holomorphic curve ϕ is unramified (cf. [18]). So by solving the polynomial equation (1.1), we can also obtain ramified holomorphic two-spheres of constant curvature in G(2, n).…”

“…[21], [22]). Later, with the totally unramified assumption, the second author, Jiao and Zhou ( [18]) found that K = 4/r, r = 3, 6 for linearly full holomorphic two-spheres with constant curvature in G(2, 5), and there exist two families of non-homogeneous holomorphic two-spheres in the case of r = 3, while there exists only a homogeneous one in the case of r = 6. So the accurate values in K h 2,5 and K f h 2,5 are still not clear.…”