Variations along the Fuchsian locus

TL;DR

This paper provides an explicit formula for the Pressure Metric on the Hitchin component along the Fuchsian locus, linking it to holomorphic differentials and classical Teichmüller theory results.

Contribution

It introduces a new explicit expression for the Pressure Metric in higher Teichmüller spaces, extending classical variational formulas to the Hitchin component.

Findings

Explicit Pressure Metric formula in terms of holomorphic differentials

Generalized variational formulas including Gardiner's formula

Established relationships between length functions and deformations

Abstract

The main result is an explicit expression for the Pressure Metric on the Hitchin component of surface group representations into PSL(n,R) along the Fuchsian locus. The expression is in terms of a parametrization of the tangent space by holomorphic differentials, and it gives a precise relationship with the Petersson pairing. Along the way, variational formulas are established that generalize results from classical Teichmueller theory, such as Gardiner's formula, the relationship between length functions and Fenchel-Nielsen deformations, and variations of cross ratios.