  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.
