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

Abstract

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.