Eigenvalues and Entropy of a Hitchin representation

TL;DR

This paper establishes an upper bound for the critical exponent of Hitchin representations in PSL(d,R), linking geometric properties of minimal surfaces and the regularity of associated Frenet curves, with equality only at Fuchsian points.

Contribution

It introduces a new inequality for the critical exponent in Hitchin components and relates the smoothness of Frenet curves to Fuchsian representations.

Findings

Critical exponent is bounded above in Hitchin representations.

Equality in the bound characterizes Fuchsian representations.

Smooth Frenet curves imply the representation is Fuchsian.

Abstract

We show that the critical exponent of a representation in the Hitchin component of $PSL(d,\mathbb{R})$ is bounded above, the least upper bound being attained only in the Fuchsian locus. This provides a rigid inequality for the area of a minimal surface on $\rho\backslash X,$ where $X$ is the symmetric space of $PSL(d,\mathbb{R}).$ The proof relies in a construction useful to prove a regularity statement: if the Frenet equivariant curve of $\rho$ is smooth, then $\rho$ is Fuchsian.