Complex Lagrangian minimal surfaces, bi-complex Higgs bundles and $\mathrm{SL}(3,\mathbb{C})$-quasi-Fuchsian representations
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929393881841664 |
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| author | Rungi, Nicholas Tamburelli, Andrea |
| author_facet | Rungi, Nicholas Tamburelli, Andrea |
| contents | In this paper we introduce complex minimal Lagrangian surfaces in the bi-complex hyperbolic space and study their relation with representations in $\mathrm{SL}(3,\mathbb{C})$. Our theory generalizes at the same time minimal Lagrangian surfaces in the complex hyperbolic plane, hyperbolic affine spheres in $\mathbb{R}^3$, and Bers embeddings in the holomorphic space form $\mathbb{CP}^1 \times \mathbb{CP}^1 \setminus Δ$. If these surfaces are equivariant under representations in $\mathrm{SL}(3,\mathbb{C})$, our approach generalizes the study of almost $\mathbb{R}$-Fuchsian representations in $\mathrm{SU}(2,1)$, Hitchin representations in $\mathrm{SL}(3,\mathbb{R})$, and quasi-Fuchsian representations in $\mathrm{SL}(2,\mathbb{C})$. Moreover, we give a parameterization of $\mathrm{SL}(3,\mathbb{C})$-quasi-Fuchsian representations by an open set in the product of two copies of the bundle of holomorphic cubic differentials over the Teichmüller space of $S$, from which we deduce that this space of representations is endowed with a bi-complex structure. In the process, we introduce bi-complex Higgs bundles as a new tool for studying representations into semisimple complex Lie groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_14945 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Complex Lagrangian minimal surfaces, bi-complex Higgs bundles and $\mathrm{SL}(3,\mathbb{C})$-quasi-Fuchsian representations Rungi, Nicholas Tamburelli, Andrea Differential Geometry Geometric Topology Symplectic Geometry In this paper we introduce complex minimal Lagrangian surfaces in the bi-complex hyperbolic space and study their relation with representations in $\mathrm{SL}(3,\mathbb{C})$. Our theory generalizes at the same time minimal Lagrangian surfaces in the complex hyperbolic plane, hyperbolic affine spheres in $\mathbb{R}^3$, and Bers embeddings in the holomorphic space form $\mathbb{CP}^1 \times \mathbb{CP}^1 \setminus Δ$. If these surfaces are equivariant under representations in $\mathrm{SL}(3,\mathbb{C})$, our approach generalizes the study of almost $\mathbb{R}$-Fuchsian representations in $\mathrm{SU}(2,1)$, Hitchin representations in $\mathrm{SL}(3,\mathbb{R})$, and quasi-Fuchsian representations in $\mathrm{SL}(2,\mathbb{C})$. Moreover, we give a parameterization of $\mathrm{SL}(3,\mathbb{C})$-quasi-Fuchsian representations by an open set in the product of two copies of the bundle of holomorphic cubic differentials over the Teichmüller space of $S$, from which we deduce that this space of representations is endowed with a bi-complex structure. In the process, we introduce bi-complex Higgs bundles as a new tool for studying representations into semisimple complex Lie groups. |
| title | Complex Lagrangian minimal surfaces, bi-complex Higgs bundles and $\mathrm{SL}(3,\mathbb{C})$-quasi-Fuchsian representations |
| topic | Differential Geometry Geometric Topology Symplectic Geometry |
| url | https://arxiv.org/abs/2406.14945 |