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arXiv · 1405.4748

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

Max Bauer·Elise Goujard
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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.

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Max Bauer, Elise Goujard. 2014-09-28. Geometry of periodic regions on flat surfaces and associated Siegel-Veech constants. https://arxiv.org/abs/1405.4748

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