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- Title
Non-Strict Plurisubharmonicity of Energy on Teichmüller Space.
- Authors
Tošić, Ognjen
- Abstract
For an irreducible representation |$\rho :\pi _{1}(\Sigma _{g})\to \textrm{GL}(n,\mathbb{C})$| , there is an energy functional |$\textrm{E}_{\rho }: {{\mathcal{T}}}_{g}\to \mathbb{R}$| , defined on Teichmüller space by taking the energy of the associated equivariant harmonic map into the symmetric space |$\textrm{GL}(n,\mathbb{C})/\textrm{U}(n)$|. It follows from a result of Toledo that |$\textrm{E}_{\rho }$| is plurisubharmonic, that is, its Levi form is positive semi-definite. We describe 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,\rho)$| , the map |$\textrm{E}_{\rho }$| is strictly plurisubharmonic. We also describe the kernel of the Levi form for |$n=1$|.
- Subjects
TOLEDO (Ohio); TEICHMULLER spaces; HARMONIC maps; SYMMETRIC spaces
- Publication
IMRN: International Mathematics Research Notices, 2024, Vol 2024, Issue 9, p7820
- ISSN
1073-7928
- Publication type
Article
- DOI
10.1093/imrn/rnad325