Structure of minimal 2-spheres of constant curvature in the complex hyperquadric

TL;DR

This paper investigates the structure and classification of minimal 2-spheres with constant curvature in complex hyperquadrics and projective spaces, introducing a moduli space framework and constructing various examples.

Contribution

It develops a moduli space description for minimal 2-spheres of constant curvature using complex symmetric matrices and explores their uniqueness and explicit examples.

Findings

Moduli space characterized by complex symmetric matrices

Construction of higher degree holomorphic 2-spheres

Proof of uniqueness for totally real minimal 2-spheres

Abstract

In this paper, the singular-value decomposition theory of complex matrices is explored to study constantly curved 2-spheres minimal in both $\mathbb{C}P^n$ and the hyperquadric of $\mathbb{C}P^n$. The moduli space of all those noncongruent ones is introduced, which can be described by certain complex symmetric matrices modulo an appropriate group action. Using this description, many examples, such as constantly curved holomorphic 2-spheres of higher degree, nonhomogenous minimal 2-spheres of constant curvature, etc., are constructed. Uniqueness is proven for the totally real constantly curved 2-sphere minimal in both the hyperquadric and $\mathbb{C}P^n$.