Siegel Veech Constants Research Articles

Euler characteristics of Teichmüller curves in genus two

We calculate the Euler characteristics of all of the Teichmuller curves in the moduli space of genus two Riemann surfaces which are generated by holomorphic one-forms with a single double zero. These curves can all be embedded in Hilbert modular surfaces and our main result is that the Euler characteristic of a Teichmuller curve is proportional to the Euler characteristic of the Hilbert modular surface on which it lies. The idea is to use techniques from algebraic geometry to calculate the fundamental classes of these Teichmuller curves in certain compactifications of the Hilbert modular surfaces. This is done by defining meromorphic sections of line bundles over Hilbert modular surfaces which vanish along these Teichmuller curves. We apply these results to calculate the Siegel-Veech constants for counting closed billiards paths in certain L-shaped polygons. We also calculate the Lyapunov exponents of the Kontsevich-Zorich cocycle for any ergodic, SL_2(R)-invariant measure on the moduli space of Abelian differentials in genus two (previously calculated in unpublished work of Kontsevich and Zorich).

Siegel–Veech constants in ℋ(2)

Abelian differentials on Riemann surfaces can be seen as translation surfaces, which are flat surfaces with cone-type singularities. Closed geodesics for the associated flat metrics form cylinders whose number under a given maximal length generically has quadratic asymptotics in this length, with a common coefficient constant for the quadratic asymptotics called a Siegel--Veech constant which is shared by almost all surfaces in each moduli space of translation surfaces. Square-tiled surfaces are specific translation surfaces which have their own quadratic asymptotics for the number of cylinders of closed geodesics. It is an interesting question whether, as n tends to infinity, the Siegel--Veech constants of square-tiled surfaces with n tiles tend to the generic constants of the ambient moduli space. We prove that this is the case in the moduli space H(2) of translation surfaces of genus two with one singularity.

Moduli spaces of Abelian differentials: The principal boundary, counting problems, and the Siegel–Veech constants

A holomorphic 1-form on a compact Riemann surface S naturally defines a flat metric on S with cone-type singularities. We present the following surprising phenomenon: having found a geodesic segment (saddle connection) joining a pair of conical points one can find with a nonzero probability another saddle connection on S having the same direction and the same length as the initial one. A similar phenomenon is valid for the families of parallel closed geodesics.We give a complete description of all possible configurations of parallel saddle connections (and of families of parallel closed geodesics) which might be found on a generic flat surface S. We count the number of saddle connections of length less than L on a generic flat surface S; we also count the number of admissible configurations of pairs (triples,...) of saddle connections; we count the analogous numbers of configurations of families of closed geodesics. By the previous result of [EMa] these numbers have quadratic asymptotics c·(πL2). Here we explicitly compute the constant c for a configuration of every type. The constant c is found from a Siegel–Veech formula.To perform this computation we elaborate the detailed description of the principal part of the boundary of the moduli space of holomorphic 1-forms and we find the numerical value of the normalized volume of the tubular neighborhood of the boundary. We use this for evaluation of integrals over the moduli space.