Lower bounds for Lyapunov exponents of flat bundles on curves

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 than or equal to the degree of any rank-$k$ holomorphic subbundle. We generalize the original context from Teichmüller curves to any local system over a curve with nonexpanding cusp monodromies. As an application we obtain the large-genus limits of individual Lyapunov exponents in hyperelliptic strata of abelian differentials, which Fei Yu proved conditionally on his conjecture.

Understanding the case of equality with the degrees of subbundle coming from the Hodge filtration seems challenging, eg 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.