# Harmonic surfaces in the Cayley plane

## TL;DR

This paper explores the twistor theory of harmonic maps from Riemann surfaces into the Cayley plane, employing Lie algebra classifications to develop new geometric constructions.

## Contribution

It introduces a novel approach using nilpotent orbit classification to study harmonic maps into the Cayley plane via twistor methods.

## Key findings

- Established a twistor construction for harmonic maps into the Cayley plane.
- Linked Lie algebra nilpotent orbit classification with geometric harmonic map theory.
- Extended techniques from classical Grassmannian cases to the Cayley plane.

## Abstract

We consider the twistor theory of nilconformal harmonic maps from a Riemann surface into the Cayley plane $\mathbf{O} P^2=F_4/\mathrm{Spin}(9)$. By exhibiting this symmetric space as a submanifold of the Grassmannian of $10$-dimensional subspaces of the fundamental representation of $F_4$, techniques and constructions similar to those used in earlier works on twistor constructions of nilconformal harmonic maps into classical Grassmannians can also be applied in this case. The originality of our approach lies on the use of the classification of nilpotent orbits in Lie algebras as described by D. Djokovi\'{c}.