Non-strict plurisubharmonicity of energy on Teichmüller space
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929230472806400 |
|---|---|
| author | Tošić, Ognjen |
| author_facet | Tošić, Ognjen |
| contents | For an irreducible representation $ρ:π_1(Σ_g)\to\mathrm{GL}(n,\mathbb{C})$ there is an energy functional $\mathrm{E}_ρ:\mathcal{T}_g\to\mathbb{R}$, defined on Teichmüller space by taking the energy of the associated equivariant harmonic map into the symmetric space $\mathrm{GL}(n,\mathbb{C})/\mathrm{U}(n)$. It follows from a result of Toledo that $\mathrm{E}_ρ$ is plurisubharmonic, i.e. its Levi form is positive semi-definite. We study the kernel of this Levi form, and relate it to the $\mathbb{C}^*$ action on the moduli space of Higgs bundles. We also show that the points in $\mathcal{T}_g$ where strict plurisubharmonicity fails (i.e. this kernel is non-zero) are critical points of the Hitchin fibration. When $n\geq 2$ and $g\geq 3$, we show that for a generic choice $(S,ρ)$, the map $\mathrm{E}_ρ$ is strictly plurisubharmonic. We also describe the kernel of the Levi form for $n=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_05626 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Non-strict plurisubharmonicity of energy on Teichmüller space Tošić, Ognjen Complex Variables Differential Geometry 53C07, 53C43 For an irreducible representation $ρ:π_1(Σ_g)\to\mathrm{GL}(n,\mathbb{C})$ there is an energy functional $\mathrm{E}_ρ:\mathcal{T}_g\to\mathbb{R}$, defined on Teichmüller space by taking the energy of the associated equivariant harmonic map into the symmetric space $\mathrm{GL}(n,\mathbb{C})/\mathrm{U}(n)$. It follows from a result of Toledo that $\mathrm{E}_ρ$ is plurisubharmonic, i.e. its Levi form is positive semi-definite. We study the kernel of this Levi form, and relate it to the $\mathbb{C}^*$ action on the moduli space of Higgs bundles. We also show that the points in $\mathcal{T}_g$ where strict plurisubharmonicity fails (i.e. this kernel is non-zero) are critical points of the Hitchin fibration. When $n\geq 2$ and $g\geq 3$, we show that for a generic choice $(S,ρ)$, the map $\mathrm{E}_ρ$ is strictly plurisubharmonic. We also describe the kernel of the Levi form for $n=1$. |
| title | Non-strict plurisubharmonicity of energy on Teichmüller space |
| topic | Complex Variables Differential Geometry 53C07, 53C43 |
| url | https://arxiv.org/abs/2305.05626 |