Rigidity of conformal minimal immersions of constant curvature from $S^2$ to $Q_4$

TL;DR

This paper investigates the geometry of conformal minimal two-spheres in a complex hyperquadric, constructing examples and classifying all such immersions of constant curvature, revealing their non-congruence in general.

Contribution

It constructs a non-homogeneous minimal two-sphere in $Q_4$ and classifies all linearly full conformal minimal immersions of constant curvature from $S^2$ to $Q_4$, showing they are generally not congruent.

Findings

Existence of a non-homogeneous constant curvature minimal two-sphere in $Q_4$

Complete classification of conformal minimal immersions of constant curvature from $S^2$ to $Q_4$

Minimal two-spheres of constant curvature are generally not congruent

Abstract

Geometry of conformal minimal two-spheres immersed in $G(2,6;\mathbb{R})$ is studied in this paper by harmonic maps. We construct a non-homogeneous constant curved minimal two-sphere in $G(2,6;\mathbb{R})$, and give a classification theorem of linearly full conformal minimal immersions of constant curvature from $S^2$ to $G(2,6;\mathbb{R})$, or equivalently, a complex hyperquadric $Q_{4}$, which illustrates minimal two-spheres of constant curvature in $Q_{4}$ are in general not congruent.