Superminimal Surfaces in the 6-Sphere

TL;DR

This paper constructs explicit examples of almost complex 2-spheres in the 6-sphere with limited singularities using harmonic sequences, analyzes their symmetry properties, and characterizes their moduli space and equivalence classes.

Contribution

It introduces a method to explicitly construct and classify almost complex 2-spheres in S^6 with at most two singularities, revealing their symmetry and moduli space structure.

Findings

Explicit examples of almost complex 2-spheres with at most two singularities

Identification of symmetry properties of these spheres

Characterization of projectively equivalent curves via G_2^C-classes

Abstract

In this article, we use the harmonic sequence associated to a weakly conformal harmonic map $f:S\to S^6$ in order to determine explicit examples of linearly full almost complex 2-spheres of $S^6$ with at most two singularities. We prove that the singularity type of these almost complex 2-spheres has an extra symmetry and this allows us to determine the moduli space of such curves with suitably small area. We also characterize projectively equivalent almost complex curves of $S^6$ in terms of $G_2^{\cc}$-equivalence of their directrix curves.