Quadratic differentials in Teichmüller theory and complex harmonic maps
Abstract
We show that Hitchin-Wolf's harmonic map parametrization of Teichmüller space, the Schwarzian holonomy map, and Uhlenbeck's almost-Fuchsian holonomy map all arise as restrictions of a single locally biholomorphic map into the PSL(2,ℂ) character variety. The map admits a right inverse defined on an open subset of quasi-Fuchsian space, encoding harmonic maps together with the Bers embedding. The construction goes through complex harmonic maps to the space of oriented geodesics of ℍ3. Along the way, we further develop the theory of complex harmonic maps, with a focus on the natural ℂ*-action and the relation to complex projective structures. This leads to an independent application: we obtain a new geometric interpretation of certain real twistor lines in the rank-two Deligne-Hitchin moduli space and we recover a theorem of Gothen-Silva.
Related
- Quadratic differentials and random walks on the dual graph of a pants decomposition
- Asymptotics of Masur-Veech volume of strata of quadratic differentials with many simple poles
- Perverse schobers, stability conditions and quadratic differentials I
- Harmonic Quasiconformal Mappings in the BMO-Teichmüller space
- Totally Geodesic Submanifolds of Teichmüller Space in Complex Dimension at Least Three
BibTeX
@misc{emam2026quadratic,
title = {{Quadratic differentials in Teichmüller theory and complex harmonic maps}},
author = {Christian El Emam and Nathaniel Sagman},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08885},
eprint = {2610.08885},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08885},
note = {CC0},
}
RIS
TY - UNPB
TI - Quadratic differentials in Teichmüller theory and complex harmonic maps
AU - Christian El Emam
AU - Nathaniel Sagman
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08885
UR - https://arxiv.org/abs/2610.08885
AB - We show that Hitchin-Wolf's harmonic map parametrization of Teichmüller space, the Schwarzian holonomy map, and Uhlenbeck's almost-Fuchsian holonomy map all arise as restrictions of a single locally biholomorphic map into the PSL(2,ℂ) character variety. The map admits a right inverse defined on an open subset of quasi-Fuchsian space, encoding harmonic maps together with the Bers embedding. The construction goes through complex harmonic maps to the space of oriented geodesics of ℍ^3. Along the way, we further develop the theory of complex harmonic maps, with a focus on the natural ℂ^*-action and the relation to complex projective structures. This leads to an independent application: we obtain a new geometric interpretation of certain real twistor lines in the rank-two Deligne-Hitchin moduli space and we recover a theorem of Gothen-Silva.
ER -