Complex harmonic maps and rank 2 higher Teichmüller theory
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916793378930688 |
|---|---|
| author | Emam, Christian El Sagman, Nathaniel |
| author_facet | Emam, Christian El Sagman, Nathaniel |
| contents | 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 Teichmüller 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 Teichmüller 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-Kähler structure. We obtain partial generalizations in higher rank, and we construct Kähler structures on other spaces that are related to the Hitchin components. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_11746 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Complex harmonic maps and rank 2 higher Teichmüller theory Emam, Christian El Sagman, Nathaniel Differential Geometry Complex Variables Geometric Topology Representation Theory 58E20, 20H10, 30F60 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 Teichmüller 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 Teichmüller 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-Kähler structure. We obtain partial generalizations in higher rank, and we construct Kähler structures on other spaces that are related to the Hitchin components. |
| title | Complex harmonic maps and rank 2 higher Teichmüller theory |
| topic | Differential Geometry Complex Variables Geometric Topology Representation Theory 58E20, 20H10, 30F60 |
| url | https://arxiv.org/abs/2506.11746 |