Geodesic Coordinates for the Pressure Metric at the Fuchsian Locus

TL;DR

This paper demonstrates that the Hitchin parametrization provides natural geodesic coordinates for the pressure metric at the Fuchsian locus in the Hitchin component for surface group representations into PSL(3,R), using thermodynamic formalism and gauge theory.

Contribution

It establishes that the Hitchin parametrization yields geodesic coordinates for the pressure metric at the Fuchsian locus in the Hitchin component, with detailed derivative computations.

Findings

First derivatives of the pressure metric vanish at the Fuchsian locus.

Derived explicit formulas for first and second variations of reparametrization functions.

Extended surface integrals showing cancellations due to symmetries.

Abstract

We prove that the Hitchin parametrization provides geodesic coordinates at the Fuchsian locus for the pressure metric in the Hitchin component $\mathcal{H}_{3}(S)$ of surface group representations into $PSL(3,\mathbb{R})$. The proof consists of the following elements: we compute first derivatives of the pressure metric using the thermodynamic formalism. We invoke a gauge-theoretic formula to compute first and second variations of reparametrization functions by studying flat connections from Hitchin's equations and their parallel transports. We then extend these expressions of integrals over closed geodesics to integrals over the two-dimensional surface. Symmetries of the Liouville measure then provide cancellations, which show that the first derivatives of the pressure metric tensors vanish at the Fuchsian locus.