Non-Strict Plurisubharmonicity of Energy on Teichmüller Space

Abstract

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 $n=1$.