Geodesic coordinates for the pressure metric at the Fuchsian locus

Dai, Xian

doi:10.2140/gt.2023.27.1391

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Geodesic coordinates for the pressure metric at the Fuchsian locus

Xian Dai

Geometry & Topology 27 (2023) 1391–1478

DOI: 10.2140/gt.2023.27.1391

Abstract

We prove that the Hitchin parametrization provides geodesic coordinates at the Fuchsian locus for the pressure metric in the Hitchin component $ℋ_{3} (S)$ of surface group representations into $PSL (3, ℝ)$ .

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 the first and second variations of the 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.

Keywords

pressure metric, Hitchin representations, Higgs bundles, thermodynamic formalism

Mathematical Subject Classification 2010

Primary: 53B20

Secondary: 37D35

References

Publication

Received: 24 January 2020

Revised: 15 July 2021

Accepted: 4 October 2021

Published: 15 June 2023

Proposed: Anna Wienhard

Seconded: Paul Seidel, Leonid Polterovich

Authors

Xian Dai
Heidelberg University Heidelberg Germany

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Volume 27, issue 4 (2023)