Lower bounds for Lyapunov exponents of flat bundles on curves

TL;DR

This paper proves a lower bound for the sum of top Lyapunov exponents of flat bundles over curves, extending previous conjectures and exploring implications for hyperelliptic strata and monodromy groups.

Contribution

It generalizes a conjecture to all local systems with non-expanding cusp monodromies and establishes new lower bounds for Lyapunov exponents on curves.

Findings

Proved the conjecture of Fei Yu on Lyapunov exponents.

Derived large genus limits of Lyapunov exponents in hyperelliptic strata.

Suggested a link between equality cases and thin monodromy groups.

Abstract

Consider a flat bundle over a complex curve. We prove a conjecture of Fei Yu that the sum of the top k Lyapunov exponents of the flat bundle is always greater or equal to the degree of any rank k holomorphic subbundle. We generalize the original context from Teichmueller curves to any local system over a curve with non-expanding cusp monodromies. As an application we obtain the large genus limits of individual Lyapunov exponents in hyperelliptic strata of Abelian differentials. Understanding the case of equality with the degrees of subbundle coming from the Hodge filtration seems challenging, e.g. for Calabi-Yau type families. We conjecture that equality of the sum of Lyapunov exponents and the degree is related to the monodromy group being a thin subgroup of its Zariski closure.