Siegel-Veech Measures of Convex Flat Cone Spheres

TL;DR

This work extends Siegel–Veech-type counting from translation surfaces to convex flat cone spheres with positive curvature gap, constructing generalized Siegel–Veech transforms and associated measures by integrating over the Thurston moduli space with the complex hyperbolic metric. It proves the transforms are bounded, defines absolutely continuous Siegel–Veech measures, and shows these measures are piecewise real analytic; it also derives small-length asymptotics expressed via boundary-stratum data and explicit leading coefficients. The methodology combines spanning-tree/Delaunay coordinate charts, o-minimal semialgebraicity, Thurston surgeries, and a product-metric approximation to obtain precise asymptotics for volumes of thin parts of moduli spaces. The results generalize the classical Siegel–Veech formula to irrational-curvature flat cones, yielding analogues of Siegel–Veech constants and sharpening the understanding of geodesic statistics on convex flat cone spheres with positive curvature gap.

Abstract

A classical theorem of Siegel gives the average number of lattice points in bounded subsets of $\mathbb{R}^n$. Motivated by this result, Veech introduced an analogue for translation surfaces, known as the Siegel-Veech formula, which describes the average number of saddle connections of bounded length on the moduli space of translation surfaces. However, no such formula is known for flat surfaces with cone angles that are irrational multiples of $π$. A convex flat cone sphere is a Riemann sphere equipped with a conformal flat metric with conical singularities, all of whose cone angles lie in the interval $(0, 2π)$. In this paper, we extend the Siegel-Veech formula to this setting. We define a generalized Siegel-Veech transform and prove that it belongs to $L^\infty$ on the moduli space. This leads to the definition of a Siegel-Veech measure on $\mathbb{R}_{>0}$, obtained by integrating the Siegel-Veech transform over the moduli space. This measure can be viewed as a generalization of the classical Siegel-Veech formula. We show that it is absolutely continuous and piecewise real analytic. Finally, we study the asymptotic behavior of this measure on small intervals $(0,\varepsilon)$ as $\varepsilon \to 0$, providing an analogue of Siegel-Veech constants for convex flat cone spheres.

Figures (8)

• Figure 2.1: Thurston surgery glues $C_{\gamma}$ on the right along the slit saddle connection $\gamma$ in the left flat cone sphere.
• Figure 2.2: The gray regions on the left form a convex hull $D_1$ in $X$; the generalized Thurston surgery cut out $D_1$ and glue $C_{D_1}$ (on the right) to $X\setminus D_1$ along the dotted line $\partial D_1$.
• Figure 4.1: $P(\gamma, X,T)$ and the complex vector $w_{v_1}(v)$ associated to the vertices $v$ and $v_1$.
• Figure 4.2: The unfolding $P(\gamma, X, T, e_0)$ and the complex vector $w_v(v')$ associated to the vertices $v$ and $v'$. The gray region indicates the corridor of the regular closed geodesic $\gamma$.
• Figure 5.1: Shortening the sum of the lengths of $e'$ and $e_1$.

Theorems & Definitions (142)

• Definition 1.1
• Remark 1.2
• Definition 1.3: Siegel-Veech transforms
• Definition 1.4: Siegel-Veech measures
• Remark 1.5
• Theorem 1.6
• Remark 1.7
• Remark 1.8
• Remark 1.9: Comparison of Siegel-Veech transforms
• Remark 1.10: Comparison of Integrability