Geometry & Topology

Eigenvalues of curvature, Lyapunov exponents and Harder–Narasimhan filtrations

Fei Yu

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Inspired by the Katz–Mazur theorem on crystalline cohomology and by the numerical experiments of Eskin, Kontsevich and Zorich, we conjecture that the polygon of the Lyapunov spectrum lies above (or on) the Harder–Narasimhan polygon of the Hodge bundle over any Teichmüller curve. We also discuss the connections between the two polygons and the integral of eigenvalues of the curvature of the Hodge bundle by using the works of Atiyah and Bott, Forni, and Möller. We obtain several applications to Teichmüller dynamics conditional on the conjecture.

Article information

Geom. Topol., Volume 22, Number 4 (2018), 2253-2298.

Received: 12 October 2016
Accepted: 21 August 2017
First available in Project Euclid: 13 April 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14H10: Families, moduli (algebraic) 30F60: Teichmüller theory [See also 32G15] 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]
Secondary: 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 53C07: Special connections and metrics on vector bundles (Hermite-Einstein- Yang-Mills) [See also 32Q20]

moduli space of Riemann surface Teichmüller geodesic flow eigenvalue of curvature Lyapunov exponent Harder–Narasimhan filtration


Yu, Fei. Eigenvalues of curvature, Lyapunov exponents and Harder–Narasimhan filtrations. Geom. Topol. 22 (2018), no. 4, 2253--2298. doi:10.2140/gt.2018.22.2253.

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