Primitive immersions of constant curvature of surfaces into flag manifolds

TL;DR

This paper studies primitive immersions of constant curvature surfaces into flag manifolds, showing that such immersions from the sphere are equivalent to Veronese lifts and exploring invariant metrics that maximize induced area.

Contribution

It proves that primitive immersions from the sphere with constant curvature are equivalent to Veronese lifts and extends partial results to general simply connected surfaces.

Findings

Primitive immersions from the sphere are unitarily equivalent to Veronese lifts.

Such immersions have constant curvature with respect to all invariant metrics.

The paper identifies invariant metrics that maximize the induced area for primitive immersions.

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.