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| Main Authors: | , |
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| Format: | Preprint |
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2020
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2002.07295 |
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| _version_ | 1866915254017982464 |
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| author | Tamburelli, Andrea Wolf, Michael |
| author_facet | Tamburelli, Andrea Wolf, Michael |
| contents | We study the asymptotic geometry of a family of conformally planar minimal surfaces with polynomial growth in the $\mathrm{Sp}(4,\mathbb{R})$-symmetric space. We describe a homeomomorphism between the "Hitchin component" of wild $\mathrm{Sp}(4,\mathbb{R})$-Higgs bundles over $\mathbb{CP}^1$ with a single pole at infinity and a component of maximal surfaces with light-like polygonal boundary in $\mathbb{H}^{2,2}$. Moreover, we identify those surfaces with convex embeddings into the Grassmannian of symplectic planes of $\mathbb{R}^{4}$. We show, in addition, that our planar maximal surfaces are the local limits of equivariant maximal surfaces in $\mathbb{H}^{2,2}$ associated to $\mathrm{Sp}(4,\mathbb{R})$-Hitchin representations along rays of holomorphic quartic differentials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2002_07295 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Planar minimal surfaces with polynomial growth in the $\mathrm{Sp}(4, \mathbb{R})$-symmetric space Tamburelli, Andrea Wolf, Michael Differential Geometry Analysis of PDEs We study the asymptotic geometry of a family of conformally planar minimal surfaces with polynomial growth in the $\mathrm{Sp}(4,\mathbb{R})$-symmetric space. We describe a homeomomorphism between the "Hitchin component" of wild $\mathrm{Sp}(4,\mathbb{R})$-Higgs bundles over $\mathbb{CP}^1$ with a single pole at infinity and a component of maximal surfaces with light-like polygonal boundary in $\mathbb{H}^{2,2}$. Moreover, we identify those surfaces with convex embeddings into the Grassmannian of symplectic planes of $\mathbb{R}^{4}$. We show, in addition, that our planar maximal surfaces are the local limits of equivariant maximal surfaces in $\mathbb{H}^{2,2}$ associated to $\mathrm{Sp}(4,\mathbb{R})$-Hitchin representations along rays of holomorphic quartic differentials. |
| title | Planar minimal surfaces with polynomial growth in the $\mathrm{Sp}(4, \mathbb{R})$-symmetric space |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2002.07295 |