Length functions of Hitchin representations

Abstract

Given a Hitchin representation $ρ\colon π_1(S) \to \PSL_n(\mathbb{R})$, we construct $n$ continuous functions $\ell_i^ρ\colon \mathcal \CH(S) \to \mathbb{R}$ defined on the space of Hölder geodesic currents $\CH(S)$ such that, for a closed, oriented curve $γ$ in $S$, the $i$--th eigenvalue of the matrix $ρ(γ)\in \PSL_n(\mathbb{R})$ is of the form $\pm \mathrm{exp}\, \ell_i^ρ(γ)$: such functions generalize to higher rank Thurston's length function of Fuchsian re\presentations. Identities, diffe\rentiability properties of these lengths $\ell_i^ρ$, as well as applications to eigenvalue estimates, are also considered.