Triangulations and volume form on moduli spaces of flat surfaces

In this paper, we are interested in flat metric structures with conical singularities on surfaces which are obtained by deforming translation surface structures. The moduli space of such flat metric structures can be viewed as some deformation of the moduli space of translation surfaces. Using geodesic triangulations, we define a volume form on this moduli space, and show that, in the well-known cases, this volume form agrees with usual ones, up to a multiplicative constant.Comment: 42 page

Square-tiled cyclic covers

A cyclic cover of the complex projective line branched at four appropriate points has a natural structure of a square-tiled surface. We describe the combinatorics of such a square-tiled surface, the geometry of the corresponding Teichm\"uller curve, and compute the Lyapunov exponents of the determinant bundle over the Teichm\"uller curve with respect to the geodesic flow. This paper includes a new example (announced by G. Forni and C. Matheus in \cite{Forni:Matheus}) of a Teichm\"uller curve of a square-tiled cyclic cover in a stratum of Abelian differentials in genus four with a maximally degenerate Kontsevich--Zorich spectrum (the only known example found previously by Forni in genus three also corresponds to a square-tiled cyclic cover \cite{ForniSurvey}). We present several new examples of Teichm\"uller curves in strata of holomorphic and meromorphic quadratic differentials with maximally degenerate Kontsevich--Zorich spectrum. Presumably, these examples cover all possible Teichm\"uller curves with maximally degenerate spectrum. We prove that this is indeed the case within the class of square-tiled cyclic covers.Comment: 34 pages, 6 figures. Final version incorporating referees comments. In particular, a gap in the previous version was corrected. This file uses the journal's class file (jmd.cls), so that it is very similar to published versio

Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmuller geodesic flow

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 hyperbolic Laplacians when the underlying Riemann surface degenerates.Comment: Minor corrections. To appear in Publications mathematiques de l'IHE

Ergodic infinite group extensions of geodesic flows on translation surfaces

We show that generic infinite group extensions of geodesic flows on square tiled translation surfaces are ergodic in almost every direction, subject to certain natural constraints. Recently K. Fr\c{a}czek and C. Ulcigrai have shown that certain concrete staircases, covers of square-tiled surfaces, are not ergodic in almost every direction. In contrast we show the almost sure ergodicity of other concrete staircases. An appendix provides a combinatorial approach for the study of square-tiled surfaces

Preventive Effects of Omega-3 and Omega-6 Fatty Acids on Peroxide Mediated Oxidative Stress Responses in Primary Human Trabecular Meshwork Cells

Pathologic processes in glaucoma include increased apoptosis, accumulation of extracellular material in the trabecular meshwork and optic nerve, condensations of the cytoskeleton and precocious cellular senescence. Oxidative stress was shown to generate these alterations in primary ocular cells. Fatty acids omega-3 and -6 are alleged to constitute a prophylaxis against these deleterious effects. Here, we tested actual preventive effects omega-3 and -6 against peroxide induced stress responses in primary human trabecular meshwork cells. Changes of mitochondrial activity, proliferation, heat shock proteins, extracellular matrix components, and inflammatory markers were evaluated. Alterations of the cytoskeleton were evaluated by phalloidin labeling. Here we report a repressive effect of omega-6 on metabolic activity and proliferation, which was not detected for omega-3. Both agents were able to prevent the anti-proliferative effect of H2O2, but only omega-3 prevented metabolic repression. Expression of heat shock protein 27 was unaltered by both fatty acids, whereas heat shock protein 90 was significantly induced by both. Omega-6 increased fibronectin and connective tissue growth factor synthesis, as well as the amount of secreted fibronectin. Omega-3, instead, induced plasminogen activator inhibitor 1 synthesis. H2O2 further increased fibronectin production in omega-6 supplemented cells, which was not the case in omega-3 treated cells. H2O2 stimulation of plasminogen activator inhibitor 1 and connective tissue growth factor was repressed by both fatty acids. Both fatty acids appeared to abolish H2O2 mediated stimulation of nuclear factor κB and IL-6, but not IL-1α and IL-8. H2O2 induced formation of cross-linked actin networks and stress fibers, which was reduced by preemptive application of omega-3. Omega-6, in contrast, had no protective effect on that, and even seemed to promote condensation. Based on the observed side effects of omega-6, omega-3 appears to be the more beneficial fatty acid in respect of prophylactic intake for prevention of a glaucomatous disease