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

Abstract

Let Σ \Sigma be a closed surface of genus at least two and ρ : π 1 ( Σ ) → G \rho \colon \pi _1(\Sigma ) \to G a Hitchin representation into G = P S L ( n , R ) G=PSL(n,\mathbb {R}) , P S p ( 2 n , R ) PSp(2n,\mathbb {R}) , P S O ( n , n + 1 ) PSO(n,n+1) or G 2 G_2 . We consider the energy functional E E on the Teichmüller space of Σ \Sigma which assigns to each point in T ( Σ ) \mathcal {T}(\Sigma ) the energy of the associated ρ \rho -equivariant harmonic map. The main result of this paper is that E E is strictly plurisubharmonic. As a corollary we obtain an upper bound of 3 ⋅ genus ( Σ ) − 3 3 \cdot \text {genus}(\Sigma ) -3 on the index of any critical point of the energy functional.