Algebraic Harmonic Maps, Totally Symmetric Harmonic Maps and a Conjecture

TL;DR

This paper explores algebraic and totally symmetric harmonic maps from Riemann surfaces into symmetric spaces, compares these notions, and proposes a related conjecture.

Contribution

It introduces a comparison between algebraic and totally symmetric harmonic maps and formulates a new conjecture in the context of harmonic maps of finite uniton number.

Findings

Comparison of algebraic and totally symmetric harmonic maps

Formulation of a new conjecture relating these notions

Insights into harmonic maps of finite uniton number

Abstract

In this paper, we discuss the associated family of harmonic maps $\mathcal{F}: M \rightarrow G/K$ from a Riemann surface $M$ into inner symmetric spaces of compact or non-compact type which are either algebraic or totally symmetric. These notions are the two components of the definition of a harmonic map of finite uniton number, as stated by [2]. We finish this paper by comparing these to notions and by stating a conjecture.