Non-strict plurisubharmonicity of energy on Teichmüller space

Abstract

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 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 ${\mathrm{E}}_{\rho }$ is plurisubharmonic, i.e. its Levi form is positive semi-definite. We study the kernel of this Levi form, and relate it to the ${\mathbb{C}}^{\ast }$ 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 $n\ge 2$ and $g\ge 3$, 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 $n=1$.

29 pages, 0 figures

References in corpus (1)

• Unstable minimal surfaces in symmetric spaces of non-compact type