Constant curvature solutions of Grassmannian sigma models. I: Holomorphic solutions (Q1947587)

The authors consider the problem of finding holomorphic maps from \(S^2\) into a Grassmannian manifold \(G(m,n)\) with constant Gaussian curvature \(K\). They present a procedure for constructing such maps which lead them to conjecture that: a) the minimal value for \(K\) that can be obtained is the curvature \(K = {{4}\over{m(n-m)}}\) of the Veronese holomorphic curve; b) there exist constant curvature holomorphic maps from \(S^2\) into \(G(m,n)\) with \(K = {{4}\over{r}}\) for all \(1 \leq r \leq m(n-m)\). In support of their conjectures, they show that (a) and (b) are true when \(G(m, n) \) is \(G(2,3)\), \(G(2,4)\) or \(G(2,5)\) and examine the classification problem of the constant curvature holomorphic maps into \(G(2, n)\), \(n\geq 6\), making various remarks.