Non-strict plurisubharmonicity of energy on Teichm\"uller space

For an irreducible representation $\rho:\pi_1(\Sigma_g)\to\mathrm{GL}(n,\mathbb{C})$ there is an energy functional $\mathrm{E}_\rho:\mathcal{T}_g\to\mathbb{R}$, defined on Teichm\"uller 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 $\mathrm{E}_\rho$ 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 $(S,\rho)$, the map $\mathrm{E}_\rho$ is strictly plurisubharmonic. We also describe the kernel of the Levi form for $n=1$.