Surfaces in $\mathbb{R}^7$ obtained from harmonic maps in $S^6$

TL;DR

This paper explores the geometric properties of surfaces in seven-dimensional space derived from harmonic maps from Riemann surfaces into the six-sphere, classifying special types of these surfaces using harmonic sequence methods.

Contribution

It characterizes conformal harmonic immersions into S^6 whose associated surfaces in R^7 belong to specific classes like minimal, pseudo-umbilical, or isotropic surfaces, using harmonic sequence techniques.

Findings

Characterization of harmonic immersions leading to minimal surfaces in hyperspheres

Identification of conditions for surfaces with parallel mean curvature vector

Classification of pseudo-umbilical and isotropic surfaces in R^7

Abstract

We will investigate the local geometry of the surfaces in the $7$-dimensional Euclidean space associated to harmonic maps from a Riemann surface $\Sigma$ into $S^6$. By applying methods based on the use of harmonic sequences, we will characterize the conformal harmonic immersions $\varphi:\Sigma\to S^6$ whose associated immersions $F:\Sigma\to \mathbb{R}^7$ belong to certain remarkable classes of surfaces, namely: minimal surfaces in hyperspheres; surfaces with parallel mean curvature vector field; pseudo-umbilical surfaces; isotropic surfaces.