Given integers$g,n\geqslant 0$satisfying$2-2g-n<0$, let${\mathcal{M}}_{g,n}$be the moduli space of connected, oriented, complete, finite area hyperbolic surfaces of genus$g$with$n$cusps. We study the global behavior of the Mirzakhani function$B:{\mathcal{M}}_{g,n}\rightarrow \mathbf{R}_{{\geqslant}0}$which assigns to$X\in {\mathcal{M}}_{g,n}$the Thurston measure of the set of measured geodesic laminations on$X$of hyperbolic length${\leqslant}1$. We improve bounds of Mirzakhani describing the behavior of this function near the cusp of${\mathcal{M}}_{g,n}$and deduce that$B$is square-integrable with respect to the Weil–Petersson volume form. We relate this knowledge of$B$to statistics of counting problems for simple closed hyperbolic geodesics.

中文翻译：

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双曲面上简单闭合测地线的 MIRZAKHANI 函数和统计量的平方可积性

给定整数$g,n\geqslant 0$令人满意的$2-2g-n<0$， 让${\mathcal{M}}_{g,n}$是属的连通的、有向的、完全的、有限面积的双曲曲面的模空间$g$和$n$风口浪尖。我们研究 Mirzakhani 函数的全局行为$B:{\mathcal{M}}_{g,n}\rightarrow \mathbf{R}_{{\geqslant}0}$分配给$X\in {\mathcal{M}}_{g,n}$测量的测地线叠片集的瑟斯顿测量值$X$双曲线长度${\leqslant}1$. 我们改进了 Mirzakhani 的边界，描述了该函数在${\mathcal{M}}_{g,n}$并推断出$B$是关于 Weil-Petersson 体积形式的平方可积的。我们把这些知识联系起来$B$统计简单闭合双曲测地线的计数问题。