Twistors, 4-symmetric spaces and integrable systems

TL;DR

This paper explores the connection between automorphisms of Lie algebras, integrable systems, and twistor theory, revealing new integrable geometric structures related to surfaces in symmetric spaces.

Contribution

It establishes a novel link between solutions of certain automorphism-induced integrable systems and harmonic twistor lifts of conformal surfaces, especially in four-dimensional geometries.

Findings

Solutions correspond to vertically harmonic twistor lifts.

Surfaces with holomorphic mean curvature form an integrable system.

Hamiltonian stationary Lagrangian surfaces also constitute an integrable system.

Abstract

An order four automorphism of a Lie algebra gives rise to an integrable system discussed by Terng. We show that solutions of this system may be identified with certain vertically harmonic twistor lifts of conformal maps of surfaces in a Riemannian symmetric space. Specialising to 4-dimensional target, we find that surfaces with holomorphic mean curvature in 4-dimensional spaces with constant sectional or holomorphic sectional curvatures constitute an integrable system as do Hamiltonian stationary Lagrangian surfaces in a 4-dimensional Hermitian symmetric space (this last being a result of Helein-Romon).