Complex harmonic maps and rank 2 higher Teichm\"uller theory

TL;DR

This paper develops the theory of complex harmonic maps to holomorphic Riemannian symmetric spaces and applies it to higher Teichmüller theory, especially rank 2 Hitchin components, revealing new geometric structures and relations.

Contribution

It introduces complex harmonic maps into the study of higher Teichmüller theory, generalizing Higgs bundle objects and establishing new geometric and symplectic structures for rank 2 Hitchin components.

Findings

Proved a Bers-type theorem for rank 2 Hitchin components.

Established compatibility of Goldman's symplectic form with Labourie's complex structure.

Constructed Kähler structures on related spaces.

Abstract

We initiate and develop the theory of complex harmonic maps to holomorphic Riemannian symmetric spaces, which we make use of to study complex analytic aspects of higher Teichm\"uller theory, with a focus on rank $2$ Hitchin components. Complex harmonic maps lead to various generalizations of objects from the theory of Higgs bundles; for instance, the Hitchin fibration, cyclic Higgs bundles, and the affine Toda equations. Beyond such generalizations, we also find a relation between complex harmonic maps and opers. Within the realm of higher Teichm\"uller theory, for any rank $2$ Hitchin component, we prove a Bers-type theorem, which extends and improves our previous work on $\mathrm{SL}(3,\mathbb R)$, and we prove that Goldman's symplectic form is compatible with Labourie's complex structure, so that the two determine a mapping class group invariant pseudo-K\"ahler structure. We obtain partial generalizations in higher rank, and we construct K\"ahler structures on other spaces that are related to the Hitchin components.