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Geodesic coordinates for the pressure metric at the Fuchsian locus

Xian Dai

Geometry & Topology 27 (2023) 1391–1478
Abstract

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

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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