Constant energy families of harmonic maps
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917883044429824 |
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| author | Tošić, Ognjen |
| author_facet | Tošić, Ognjen |
| contents | For a negatively curved manifold $M$ and a continuous map $ψ:Σ\to M$ from a closed surface $Σ$, we study complex submanifolds of Teichmüller space $\mathcal{S}\subset\mathcal{T}(Σ)$ such that the harmonic maps $\{h_X:X\to M\text{ for }X\in\mathcal{S}\}$ in the homotopy class of $ψ$ all have equal energy. When $M$ is real analytic with negative Hermitian sectional curvature, we show that for any such $\mathcal{S}$, there exists a closed Riemann surface $Y$, such that any $h_X$ for $X\in\mathcal{S}$ factors as a holomorphic map $ϕ_X:X\to Y$ followed by a fixed harmonic map $h:Y\to M$. This answers a question posed by both Toledo and Gromov. As a first application, we show a factorization result for harmonic maps from normal projective varieties to $M$. As a second application, we study homomorphisms from finite index subgroups of mapping class groups to $π_1(M)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_13774 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Constant energy families of harmonic maps Tošić, Ognjen Differential Geometry Complex Variables 53C43, 31B05 For a negatively curved manifold $M$ and a continuous map $ψ:Σ\to M$ from a closed surface $Σ$, we study complex submanifolds of Teichmüller space $\mathcal{S}\subset\mathcal{T}(Σ)$ such that the harmonic maps $\{h_X:X\to M\text{ for }X\in\mathcal{S}\}$ in the homotopy class of $ψ$ all have equal energy. When $M$ is real analytic with negative Hermitian sectional curvature, we show that for any such $\mathcal{S}$, there exists a closed Riemann surface $Y$, such that any $h_X$ for $X\in\mathcal{S}$ factors as a holomorphic map $ϕ_X:X\to Y$ followed by a fixed harmonic map $h:Y\to M$. This answers a question posed by both Toledo and Gromov. As a first application, we show a factorization result for harmonic maps from normal projective varieties to $M$. As a second application, we study homomorphisms from finite index subgroups of mapping class groups to $π_1(M)$. |
| title | Constant energy families of harmonic maps |
| topic | Differential Geometry Complex Variables 53C43, 31B05 |
| url | https://arxiv.org/abs/2404.13774 |