Unstable minimal surfaces in symmetric spaces of non-compact type
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866910805011726336 |
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| author | Sagman, Nathaniel Smillie, Peter |
| author_facet | Sagman, Nathaniel Smillie, Peter |
| contents | We prove that if $Σ$ is a closed surface of genus at least 3 and $G$ is a split real semisimple Lie group of rank at least $3$ acting faithfully by isometries on a symmetric space $N$, then there exists a Hitchin representation $ρ:π_1(Σ)\to G$ and a $ρ$-equivariant unstable minimal map from the universal cover of $Σ$ to $N$. This follows from a new lower bound on the index of high energy minimal maps into an arbitrary symmetric space of non-compact type. Taking $G=\mathrm{PSL}(n,\mathbb{R})$, $n\geq 4$, this disproves the Labourie conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_04885 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Unstable minimal surfaces in symmetric spaces of non-compact type Sagman, Nathaniel Smillie, Peter Differential Geometry Geometric Topology We prove that if $Σ$ is a closed surface of genus at least 3 and $G$ is a split real semisimple Lie group of rank at least $3$ acting faithfully by isometries on a symmetric space $N$, then there exists a Hitchin representation $ρ:π_1(Σ)\to G$ and a $ρ$-equivariant unstable minimal map from the universal cover of $Σ$ to $N$. This follows from a new lower bound on the index of high energy minimal maps into an arbitrary symmetric space of non-compact type. Taking $G=\mathrm{PSL}(n,\mathbb{R})$, $n\geq 4$, this disproves the Labourie conjecture. |
| title | Unstable minimal surfaces in symmetric spaces of non-compact type |
| topic | Differential Geometry Geometric Topology |
| url | https://arxiv.org/abs/2208.04885 |