Harmonic spheres in outer symmetric spaces, their canonical elements and Weierstrass type representations

TL;DR

This paper classifies harmonic two-spheres in outer symmetric spaces using Murakami's involution classification and Morse theory, providing a Weierstrass-type representation and concrete examples via meromorphic functions.

Contribution

It introduces a new classification of harmonic two-spheres in outer symmetric spaces and derives a Weierstrass-type representation for these maps.

Findings

New classification of harmonic two-spheres in outer symmetric spaces

Weierstrass-type representation for harmonic maps

Explicit examples using meromorphic functions

Abstract

Making use of Murakami's classification of outer involutions in a Lie algebra and following the Morse-theoretic approach to harmonic two-spheres in Lie groups introduced by Burstall and Guest, we obtain a new classification of harmonic two-spheres in outer symmetric spaces and a Weierstrass-type representation for such maps. Several examples of harmonic maps into classical outer symmetric spaces are given in terms of meromorphic functions on $S^2$.