Eigenvalues of Curvature, Lyapunov exponents and Harder-Narasimhan filtrations

Abstract

Inspired by Katz-Mazur theorem on crystalline cohomology and by Eskin-Kontsevich-Zorich's numerical experiments, we conjecture that the polygon of 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 Atiyah-Bott, Forni and Möller's works. We obtain several applications to Teichmüller dynamics conditional to the conjecture.

37 pages. We rewrite this paper without changing the mathematics content. arXiv admin note: text overlap with arXiv:1112.5872, arXiv:1204.1707 by other authors