Related papers: Zero Lyapunov exponents of the Hodge bundle
We show an upper bound for the sum of positive Lyapunov exponents of any Teichm\"uller curve in strata of quadratic differentials with at least one zero of large multiplicity. As a corollary, it holds for any $SL(2,\mathbb R)$-invariant…
Almost 20 years ago, the first and fourth authors found examples of SL(2,R)-invariant subbundles of Hodge bundles over Teichm\"uller curves having maximally degenerate Lyapunov spectrum. For these same surfaces, we show that a natural…
We study the Lyapunov spectrum of the Kontsevich--Zorich cocycle on $SL(2,\mathbb{R})$-invariant subbundles of the Hodge bundle over the support of a $SL(2,\mathbb{R})$-invariant probability measure on the moduli space of Abelian…
We describe the Harder--Narasimhan filtration of the Hodge bundle for Teichm\"uller curves in the non-varying strata of quadratic differentials appearing in [CM2]. Moreover, we show that the Hodge bundle on the non-varying strata away from…
We compute the sum of the positive Lyapunov exponents of the Hodge bundle with respect to the Teichmuller geodesic flow. The computation is based on the analytic Riemann-Roch Theorem and uses a comparison of determinants of flat and…
We study Lyapunov exponents for flat bundles over hyperbolic curves defined via parallel transport over the geodesic flow. We consider them as invariants on the space of Hitchin representations and show that there is a gap between any two…
We consider Lyapunov exponents for flat bundles over hyperbolic curves defined via parallel transport over the geodesic flow. We refine a lower bound obtained by Eskin, Kontsevich, Moeller and Zorich showing that the sum of the first k…
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…
A cyclic cover over the Riemann sphere branched at four points inherits a natural flat structure from the "pillow" flat structure on the basic sphere. We give an explicit formula for all individual Lyapunov exponents of the Hodge bundle…
In this note we show that the results of H. Furstenberg on the Poisson boundary of lattices of semisimple Lie groups allow to deduce simplicity properties of the Lyapunov spectrum of the Kontsevich-Zorich cocycle of Teichmueller curves in…
We classify the GL(2,R)-invariant subvarieties M in strata of Abelian differentials for which any two M-parallel cylinders have homologous core curves. This answers a question of Mirzakhani and Wright. As a corollary we show that outside of…
We show that for many strata of Abelian differentials in low genus the sum of Lyapunov exponents for the Teichmueller geodesic flow is the same for all Teichmueller curves in that stratum, hence equal to the sum of Lyapunov exponents for…
We get an upper bound of the slope of each graded quotient for the Harder-Narasimhan filtration of the Hodge bundle of a Teichm\"{u}ller curve. As an application, we show that the sum of Lyapunov exponents of a Teichm\"{u}ller curve does…
We study the relationship between the Lyapunov exponents of the geodesic flow of a closed negatively curved manifold and the geometry of the manifold. We show that if each periodic orbit of the geodesic flow has exactly one Lyapunov…
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…
We prove the Zorich-Kontsevich conjecture that the non-trivial Lyapunov exponents of the Teichm\"uller flow on (any connected component of a stratum of) the moduli space of Abelian differentials on compact Riemann surfaces are all distinct.…
Under some non-invertibility and irreducibility condition, for nilmanifold Anosov maps with one-dimensional stable bundle, we get the equivalence among the existence of invariant unstable bundle, the existence of topological conjugacy to…
Let $\Mg$ denote the moduli space of compact Riemann surfaces of genus $g$. Mumford had proved that, for each fixed genus $g$, there are isomorphisms asserting that certain higher $DET$ bundles over $\Mg$ are certain fixed…
We describe all the situations in which the Kontsevich-Zorich cocycle has zero Lyapunov exponents. Confirming a conjecture of Forni, Matheus, and Zorich, this only occurs when the cocycle satisfies additional geometric constraints. We also…
In previous work, the author fully classified orbit closures in genus three with maximally many (four) zero Lyapunov exponents of the Kontsevich-Zorich cocycle. In this paper, we prove that there are no higher dimensional orbit closures in…