Geodesic coordinates for the pressure metric at the Fuchsian locus

Dai, Xian¹

¹ Heidelberg University, Heidelberg, Germany

Geometry & topology, Tome 27 (2023) no. 4, pp. 1391-1478.

Résumé

We prove that the Hitchin parametrization provides geodesic coordinates at the Fuchsian locus for the pressure metric in the Hitchin component ${\mathcal{\mathscr{H}}}_3(S)$ of surface group representations into $\mathrm{PSL}(3,ℝ)$.

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.

Détail

DOI : 10.2140/gt.2023.27.1391

Classification : 53B20, 37D35
Keywords: pressure metric, Hitchin representations, Higgs bundles, thermodynamic formalism

Bibliographie

[1] L V Ahlfors, Some remarks on Teichmüller’s space of Riemann surfaces, Ann. of Math. 74 (1961) 171 | DOI

[2] D Baraglia, G2 geometry and integrable systems, preprint (2010)

[3] R Bowen, Markov partitions for Axiom A diffeomorphisms, Amer. J. Math. 92 (1970) 725 | DOI

[4] R Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math. 95 (1973) 429 | DOI

[5] R Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, 470, Springer (2008) | DOI

[6] R Bowen, D Ruelle, The ergodic theory of Axiom A flows, Invent. Math. 29 (1975) 181 | DOI

[7] M Bridgeman, R Canary, F Labourie, Simple length rigidity for Hitchin representations, Adv. Math. 360 (2020) | DOI

[8] M Bridgeman, R Canary, F Labourie, A Sambarino, The pressure metric for Anosov representations, Geom. Funct. Anal. 25 (2015) 1089 | DOI

[9] M Bridgeman, R Canary, F Labourie, A Sambarino, Simple root flows for Hitchin representations, Geom. Dedicata 192 (2018) 57 | DOI

[10] S Choi, W M Goldman, Convex real projective structures on closed surfaces are closed, Proc. Amer. Math. Soc. 118 (1993) 657 | DOI

[11] T Chu, The Weil–Petersson metric in the moduli space, Chinese J. Math. 4 (1976) 29

[12] D Gilbarg, N S Trudinger, Elliptic partial differential equations of second order, 224, Springer (1983) | DOI

[13] W M Goldman, Convex real projective structures on compact surfaces, J. Differential Geom. 31 (1990) 791

[14] N J Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987) 59 | DOI

[15] N J Hitchin, Lie groups and Teichmüller space, Topology 31 (1992) 449 | DOI

[16] A Katok, Entropy and closed geodesics, Ergodic Theory Dynam. Systems 2 (1982) 339 | DOI

[17] A Katok, B Hasselblatt, Introduction to the modern theory of dynamical systems, 54, Cambridge Univ. Press (1995) | DOI

[18] M Kotani, T Sunada, The pressure and higher correlations for an Anosov diffeomorphism, Ergodic Theory Dynam. Systems 21 (2001) 807 | DOI

[19] F Labourie, Anosov flows, surface groups and curves in projective space, Invent. Math. 165 (2006) 51 | DOI

[20] F Labourie, R Wentworth, Variations along the Fuchsian locus, Ann. Sci. Éc. Norm. Supér. 51 (2018) 487 | DOI

[21] Q Li, Harmonic maps for Hitchin representations, Geom. Funct. Anal. 29 (2019) 539 | DOI

[22] A N Livšic, Certain properties of the homology of Y –systems, Mat. Zametki 10 (1971) 555

[23] A N Livšic, Cohomology of dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972) 1296

[24] J Loftin, Survey on affine spheres, from: "Handbook of geometric analysis, II" (editors L Ji, P Li, R Schoen, L Simon), Adv. Lect. Math. 13, International (2010) 161

[25] H Masur, M Wolf, The Weil–Petersson isometry group, Geom. Dedicata 93 (2002) 177 | DOI

[26] C T Mcmullen, Thermodynamics, dimension and the Weil–Petersson metric, Invent. Math. 173 (2008) 365 | DOI

[27] W Parry, M Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, 187–188, Soc. Math. France (1990) 268

[28] M Pollicott, On the rate of mixing of Axiom A flows, Invent. Math. 81 (1985) 413 | DOI

[29] M Pollicott, Derivatives of topological entropy for Anosov and geodesic flows, J. Differential Geom. 39 (1994) 457

[30] M. Pollicott, Multiple mixing for hyperbolic flows, preprint (2012)

[31] R Potrie, A Sambarino, Eigenvalues and entropy of a Hitchin representation, Invent. Math. 209 (2017) 885 | DOI

[32] M Ratner, Markov partitions for Anosov flows on n–dimensional manifolds, Israel J. Math. 15 (1973) 92 | DOI

[33] A Sambarino, Quantitative properties of convex representations, Comment. Math. Helv. 89 (2014) 443 | DOI

[34] A Sambarino, The orbital counting problem for hyperconvex representations, Ann. Inst. Fourier (Grenoble) 65 (2015) 1755 | DOI

[35] C T Simpson, Constructing variations of Hodge structure using Yang–Mills theory and applications to uniformization, J. Amer. Math. Soc. 1 (1988) 867 | DOI

[36] A J Tromba, Teichmüller theory in Riemannian geometry, Birkhäuser (1992) 220 | DOI

[37] M Wolf, The Teichmüller theory of harmonic maps, J. Differential Geom. 29 (1989) 449

[38] S Wolpert, Noncompleteness of the Weil–Petersson metric for Teichmüller space, Pacific J. Math. 61 (1975) 573 | DOI

[39] S A Wolpert, Geodesic length functions and the Nielsen problem, J. Differential Geom. 25 (1987) 275