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

TL;DR

This paper studies the geometric structure of periodic regions on flat surfaces derived from Abelian differentials, focusing on Siegel-Veech constants that quantify the growth rate of these regions across different surfaces.

Contribution

It evaluates Siegel-Veech constants related to periodic cylinder configurations and explores their extremal properties within and across strata of flat surfaces.

Findings

Calculated explicit Siegel-Veech constants for various configurations.

Identified extremal configurations maximizing or minimizing these constants.

Demonstrated the universality of the growth rate coefficient across almost all surfaces.

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.