Complex affine spheres and a Bers theorem for SL(3,C)
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| Format: | Preprint |
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2024
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| _version_ | 1866915335582515200 |
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| author | Emam, Christian El Sagman, Nathaniel |
| author_facet | Emam, Christian El Sagman, Nathaniel |
| contents | For $S$ a closed surface of genus at least $2$, let $\mathrm{Hit}_3(S)$ be the Hitchin component of representations to $\mathrm{SL}(3,\mathbb{R}),$ equipped with the Labourie-Loftin complex structure. We construct a mapping class group equivariant holomorphic map from a large open subset of $\mathrm{Hit}_3(S)\times \overline{\mathrm{Hit}_3(S)}$ to the $\mathrm{SL}(3,\mathbb{C})$-character variety that restricts to the identity on the diagonal and to Bers' simultaneous uniformization on $\mathrm{T}(S)\times \overline{\mathrm{T}(S)}$. The open subset contains $\mathrm{Hit}_3(S)\times \overline{\mathrm{T}(S)}$ and $\mathrm{T}(S)\times \overline{\mathrm{Hit}_3(S)}$, and the image includes the holonomies of $\mathrm{SL}(3,\mathbb{C})$-opers.
The map is realized by associating pairs of Hitchin representations to immersions into $\mathbb{C}^3$ that we call complex affine spheres, which are equivalent to certain conformal harmonic maps into $\mathrm{SL}(3,\mathbb{C})/\mathrm{SO}(3,\mathbb{C})$ and to new objects called bi-Higgs bundles. Complex affine spheres are obtained by solving a second-order complex elliptic PDE that resembles both the Beltrami and Tzitzéica equations. To study this equation we establish analytic results that should be of independent interest. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_15287 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Complex affine spheres and a Bers theorem for SL(3,C) Emam, Christian El Sagman, Nathaniel Differential Geometry Complex Variables Geometric Topology 20H10, 30F60, 58E20 For $S$ a closed surface of genus at least $2$, let $\mathrm{Hit}_3(S)$ be the Hitchin component of representations to $\mathrm{SL}(3,\mathbb{R}),$ equipped with the Labourie-Loftin complex structure. We construct a mapping class group equivariant holomorphic map from a large open subset of $\mathrm{Hit}_3(S)\times \overline{\mathrm{Hit}_3(S)}$ to the $\mathrm{SL}(3,\mathbb{C})$-character variety that restricts to the identity on the diagonal and to Bers' simultaneous uniformization on $\mathrm{T}(S)\times \overline{\mathrm{T}(S)}$. The open subset contains $\mathrm{Hit}_3(S)\times \overline{\mathrm{T}(S)}$ and $\mathrm{T}(S)\times \overline{\mathrm{Hit}_3(S)}$, and the image includes the holonomies of $\mathrm{SL}(3,\mathbb{C})$-opers. The map is realized by associating pairs of Hitchin representations to immersions into $\mathbb{C}^3$ that we call complex affine spheres, which are equivalent to certain conformal harmonic maps into $\mathrm{SL}(3,\mathbb{C})/\mathrm{SO}(3,\mathbb{C})$ and to new objects called bi-Higgs bundles. Complex affine spheres are obtained by solving a second-order complex elliptic PDE that resembles both the Beltrami and Tzitzéica equations. To study this equation we establish analytic results that should be of independent interest. |
| title | Complex affine spheres and a Bers theorem for SL(3,C) |
| topic | Differential Geometry Complex Variables Geometric Topology 20H10, 30F60, 58E20 |
| url | https://arxiv.org/abs/2406.15287 |