Primitive immersions of constant curvature of surfaces into flag manifolds

Abstract

We investigate certain immersions of constant curvature from Riemann surfaces into flag manifolds equipped with invariant metrics, namely primitive lifts associated to pseudoholomorphic maps of surfaces into complex Grassmannians. We prove that a primitive immersion from the two-sphere into the full flag manifold which has constant curvature with respect to \emph{at least one} invariant metric is unitarily equivalent to the primitive lift of a Veronese map, hence it has constant curvature with respect to \emph{all} invariant metrics. We prove a partial generalization of this result to the case where the domain is a general simply connected Riemann surface. On the way, we consider the problem of finding the invariant metric on the flag manifold, under a certain normalization condition, that maximizes the induced area of the two-sphere by a given primitive immersion.