On entropy, regularity and rigidity for convex representations of hyperbolic manifolds

Abstract

Given a convex representation $\rho :\Gamma \rightarrow {{\mathrm{PGL}}}(d,\mathbb {R})$ of a convex cocompact group $\Gamma$ of ${{\mathrm{Isom}}}_+\mathbb {H}^k,$ we find upper bounds for the quantity $\alpha h_\rho ,$ where $h_\rho$ is the entropy of $\rho$ and $\alpha$ is the Hölder exponent of the equivariant map ${{\partial }_{\infty }}\Gamma \rightarrow \mathbb {P}(\mathbb {R}^d).$ We also give rigidity statements when the upper bound is attained. This provides an analog of Thurston’s metric for convex cocompact groups of ${{\mathrm{Isom}}}_+\mathbb {H}^k.$ We then prove that if $\rho :\pi _1\Sigma \rightarrow {{\mathrm{PSL}}}(d,\mathbb {R})$ is in the Hitchin component then $\alpha h_\rho \le 2/(d-1)$ (where $\alpha$ is the Hölder exponent of Labourie’s equivariant flag curve) with equality if and only if $\rho$ is Fuchsian.