Eigenvalues of Curvature, Lyapunov exponents and Harder-Narasimhan filtrations

TL;DR

This paper explores the relationship between Lyapunov spectra and Harder-Narasimhan filtrations in Teichmüller dynamics, proposing a conjecture supported by theoretical connections and numerical experiments.

Contribution

It introduces a conjecture linking Lyapunov and Harder-Narasimhan polygons and discusses their connections using advanced geometric and dynamical tools.

Findings

Conjecture that Lyapunov polygon lies above or on the Harder-Narasimhan polygon.

Connections established between eigenvalues of curvature and these polygons.

Applications to Teichmüller dynamics derived from the conjecture.

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 Teichm\"uller 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 M\"oller's works. We obtain several applications to Teichm\"uller dynamics conditional to the conjecture.