Transgressive Harmonic Maps and SU(1,1) Self-Duality Solutions
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866909916195717120 |
|---|---|
| author | Heller, Sebastian Schiemanowski, Lothar Weiss, Hartmut |
| author_facet | Heller, Sebastian Schiemanowski, Lothar Weiss, Hartmut |
| contents | We establish a duality between harmonic maps from Riemann surfaces to hyperbolic 3-space $\mathbb{H}^3$ and harmonic maps from Riemann surfaces to de Sitter three-space $\operatorname{dS}_3$, best viewed as a generalized Gauss map. On the gauge theoretic side, it matches SU(2) and SU(1,1) solutions of Hitchin's self-duality equations via a signature flip along an eigenline of the Higgs field. Reversing this operation typically produces singular solutions, occurring where the eigenline becomes lightlike. Motivated by explicit model examples and this singular behavior, we extend this duality to a class of transgressive harmonic maps $f:M\to \mathbb{S}^3$: these are harmonic on the hemispheres equipped with the hyperbolic metric, intersect the equator orthogonally, and have vanishing Hopf differential along the crossing set. We construct large families by gluing and analyze their regularity, and as an application obtain $τ$-real negative sections of the Deligne--Hitchin moduli space of arbitrarily large energy that are not twistor lines. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_09497 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Transgressive Harmonic Maps and SU(1,1) Self-Duality Solutions Heller, Sebastian Schiemanowski, Lothar Weiss, Hartmut Differential Geometry 53C43 We establish a duality between harmonic maps from Riemann surfaces to hyperbolic 3-space $\mathbb{H}^3$ and harmonic maps from Riemann surfaces to de Sitter three-space $\operatorname{dS}_3$, best viewed as a generalized Gauss map. On the gauge theoretic side, it matches SU(2) and SU(1,1) solutions of Hitchin's self-duality equations via a signature flip along an eigenline of the Higgs field. Reversing this operation typically produces singular solutions, occurring where the eigenline becomes lightlike. Motivated by explicit model examples and this singular behavior, we extend this duality to a class of transgressive harmonic maps $f:M\to \mathbb{S}^3$: these are harmonic on the hemispheres equipped with the hyperbolic metric, intersect the equator orthogonally, and have vanishing Hopf differential along the crossing set. We construct large families by gluing and analyze their regularity, and as an application obtain $τ$-real negative sections of the Deligne--Hitchin moduli space of arbitrarily large energy that are not twistor lines. |
| title | Transgressive Harmonic Maps and SU(1,1) Self-Duality Solutions |
| topic | Differential Geometry 53C43 |
| url | https://arxiv.org/abs/2509.09497 |