Geometry of periodic regions on flat surfaces and associated Siegel-Veech constants

Max Bauer (1) , Elise Goujard (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

An Abelian differential gives rise to a flat structure (translation surface) on the underlying Riemann surface. In some directions the directional flow on the flat surface may contain a periodic region that is made up of maximal cylinders filled by parallel geodesics of the same length. The growth rate of the number of such regions counted with weights, as a function of the length, is quadratic with a coefficient, called Siegel-Veech constant, that is shared by almost all translation surfaces in the ambient stratum. We evaluate various Siegel-Veech constants associated to the geometry of configurations of periodic cylinders and their area, and study extremal properties of such configurations in a fixed stratum and in all strata of a fixed genus.

Mots clés: Siegel-Veech constants; flat surfaces; moduli spaces of Abelian differentials; configurations of saddle connections

Domaines: Topologie géométrique [math.GT]; Systèmes dynamiques [math.DS]

Max Bauer, Elise Goujard. Geometry of periodic regions on flat surfaces and associated Siegel-Veech constants. Geometriae Dedicata, 2014, 174 (1), pp.203-233. ⟨10.1007/s10711-014-0014-z⟩. ⟨hal-00992977v2⟩