Strict plurisubharmonicity of the energy on Teichmüller space associated to Hitchin representations

Abstract

Let $\Sigma$ be a closed surface of genus least two and $\rho :{\pi }_1(\Sigma )\to G$ a Hitchin representation into $G=\text{PSL}(n,\mathbb{R})$, $\text{PSp}(2n,\mathbb{R})$, $\text{PSO}(n,n+1)$ or ${\text{G}}_2$. We consider the energy functional $E$ on the Teichmüller space of $\Sigma$ which assigns to each point in $\mathcal{T}(\Sigma )$ the energy of the associated $\rho$-equivariant harmonic map. The main result of this paper is that $E$ is strictly plurisubharmonic. As a corollary we obtain an upper bound of $3\cdot \text{genus}(\Sigma )-3$ on the index of any critical point of the energy functional.

11 pages. Peer reviewed version. Corrected typos and made small improvements to exposition. To appear in Proc. Am. Math. Soc