Strict plurisubharmonicity of the energy on Teichm\"uller space associated to Hitchin representations

TL;DR

This paper proves that the energy functional related to Hitchin representations on Teichmüller space is strictly plurisubharmonic, leading to bounds on the index of its critical points, advancing understanding of geometric structures on surfaces.

Contribution

It establishes the strict plurisubharmonicity of the energy functional for Hitchin representations, a novel result connecting harmonic map energy with complex geometry.

Findings

Energy functional is strictly plurisubharmonic.

Provides an upper bound on the index of critical points.

Links harmonic map energy to complex geometric properties.

Abstract

Let $\Sigma$ be a closed surface of genus least two and $\rho \colon \pi_1(\Sigma) \to G$ 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 Teichm\"uller 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.