Volume 22, issue 4 (2018)

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Eigenvalues of curvature, Lyapunov exponents and Harder–Narasimhan filtrations

Fei Yu

Geometry & Topology 22 (2018) 2253–2298

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.

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moduli space of Riemann surface, Teichmüller geodesic flow, eigenvalue of curvature, Lyapunov exponent, Harder–Narasimhan filtration
Mathematical Subject Classification 2010
Primary: 14H10, 30F60, 32G15
Secondary: 37D25, 53C07
Received: 12 October 2016
Accepted: 21 August 2017
Published: 5 April 2018
Proposed: Benson Farb
Seconded: Dan Abramovich, Dmitri Burago
Fei Yu
School of Mathematical Sciences
Zhejiang University