Classification of primitive immmersions of constant curvature into flag manifolds

TL;DR

This work provides a complete classification of primitive minimal immersions with constant curvature from $S^2$ into the low-dimensional flag manifolds $F_{2,1,1}$ and $F_{2,2,1}$. It develops a unified framework based on harmonic sequences, Veronese-type lifts, and invariant metrics, augmented by a generalized absolute value type function to manage curvature constraints. The main results identify explicit Veronese-based primitive curves for each flag, and show that all constant-curvature cases can be generated from Veronese primitives via homogeneous projections, addition of constants, addition of primitive maps, and shifts; several curvature formulas are provided for the resulting immersions. The findings connect classical Veronese results with modern twistorial and harmonic-map techniques, and relate to prior classifications by Li99 and Jiao, offering a systematic method to produce all constant-curvature primitive lifts in these flag settings.

Abstract

We classify primitive minimal immersions of constant curvature from the two-sphere $S^2$ into the low-dimensional flag manifolds $F_{2,1,1}$ and $F_{2,2,1}$.

Theorems & Definitions (19)

• Theorem 2.1
• Theorem 2.2
• proof
• Definition 2.3
• Lemma 2.4
• Lemma 2.5
• Theorem 3.1
• proof
• Remark 3.2
• Remark 3.3