Moduli Spaces Of Quadratic Differentials Research Articles

Configurations of saddle connections of quadratic differentials on $\mathbb{CP}^1$ and on hyperelliptic Riemann surfaces

Configurations of rigid collections of saddle connections are connected component invariants for strata of the moduli space of quadratic differentials. They have been classified for strata of Abelian differentials by Eskin, Masur and Zorich. Similar work for strata of quadratic differentials has been done by Masur and Zorich, although in that case the connected components were not distinguished. We classify the configurations for quadratic differentials on ℂℙ^1 and on hyperelliptic connected components of the moduli space of quadratic differentials. We show that, in genera greater than five, any configuration that appears in the hyperelliptic connected component of a stratum also appears in the non-hyperelliptic one. For such genera, this enables to classify the configurations that appear for each connected component of each stratum.

Multiple Saddle Connections on Flat Surfaces and the Principal Boundry of the Moduli Spaces of Quadratic Differentials

We describe typical degenerations of quadratic differentials thus describing “generic cusps” of the moduli space of meromorphic quadratic differentials with at most simple poles. The part of the boundary of the moduli space which does not arise from “generic” degenerations is often negligible in problems involving information on compactification of the moduli space. However, even for a typical degeneration one may have several short loops on the Riemann surface which shrink simultaneously. We explain this phenomenon, describe all rigid configurations of short loops, present a detailed description of analogs of desingularized stable curves arising here, and show how one can reconstruct a Riemann surface endowed with a quadratic differential which is close to a “cusp” from the corresponding point at the principal boundary.

Connected components of the strata of the moduli spaces of quadratic differentials

Dans des travaux maintenant classiques, Masur [14] et Veech [21] ont demontre independamment que le flot geodesique de Teichmuller est ergodique sur chaque composante connexe de chaque strate de l'espace des modules des differentielles quadratiques. Il devient des lors interessant d'avoir une description de ces composantes ergodiques. Veech a montre que ces strates ne sont pas necessairement connexes. Dans un article recent, Kontsevich et Zorich [8] donnent une description complete des composantes dans le cas particulier ou les differentielles quadratiques sont donnees par le carre de differentielles abeliennes. Dans cet article, nous considerons le cas complementaire. Dans un precedent article [11], nous montrions que les strates ne sont pas forcement connexes. Nous donnions une serie de strates non-connexes possedant des composantes connexes hyperelliptiques. Dans cet article, nous demontrons le theoreme general annonce dans [11]: excepte quatre cas particuliers en petits genres, les strates de l'espace des modules des differentielles quadratiques ont au plus deux composantes connexes, les cas de non-connexite etant decrits exactement par [11]: une composante est hyperelliptique, l'autre non. Notre preuve repose principalement sur une nouvelle approche des differentielles quadratiques de type Jenkins-Strebel, a savoir la notion de permutations generalisees.

Hyperelliptic components of the moduli spaces of quadratic differentials with prescribed singularities

Moduli spaces of quadratic differentials with prescribed singularities are not necessarily connected. We describe here all cases when they have a special hyperelliptic connected component. We announce the general classification theorem: up to the four exceptional cases in low dimensional stratum, any stratum of meromorphic quadratic differentials is either connected, or has exactly two connected components. In this last case, one component is hyperelliptic, the other not

Parity of the spin structure defined by a quadratic differential

According to the work of Kontsevich-Zorich, the invariant that classifies non-hyperelliptic connected components of the moduli spaces of Abelian differentials with prescribed singularities,is the parity of the spin structure. We show that for the moduli space of quadratic differentials, the spin structure is constant on every stratum where it is defined. In particular this disproves the conjecture that it classifies the non-hyperelliptic connected components of the strata of quadratic differentials with prescribed singularities. An explicit formula for the parity of the spin structure is given.

Bounded geodesics in moduli space

In the moduli space of quadratic differentials over complex structures on a surface, we construct a set of full Hausdorff dimension of points with bounded Teichmüller geodesic trajectories. The main tool is quantitative nondivergence of Teichmüller horocycles, due to Minsky and Weiss. This has an application to billiards in rational polygons.

Moduli spaces of quadratic differentials

The cotangent bundle ofJ (g, n) is a union of complex analytic subvarieties, V(π), the level sets of the function “singularity pattern” of quadratic differentials. Each V(π) is endowed with a natural affine complex structure and volume element. The latter contracts to a real analytic volume element, Μπ, on the unit hypersurface, V1(π), for the Teichmuller metric. Μπ is invariant under the pure mapping class group, γ(g, n), and a certain class of functions is proved to be Lp(Μπ), 0 <p < 1, over the moduli space V1(π)/γ (g, n). In particular, Μπ(V1(π)/γ(g, n)) < ∞, a statement which generalizes a theorem by H. Masur.