随机曲面上分离闭测地线长度的大亏格渐近性

Pub Date : 2023-01-09 DOI:10.1112/topo.12276

Xin Nie, Yunhui Wu, Yuhao Xue

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引用次数: 18

摘要

在本文中，我们研究了关于模空间Mg$\mathcal上的Weil–Petersson测度的g$g$亏格随机双曲面的基本几何量{M}_g$。我们证明了当g$g$到无穷大时，一般曲面X∈Mg$X\in\mathcal{M}_g$渐近满足：（1）X$X$的分离收缩期约为2logg$2\log g\hbox｛\it；｝$（2）在X上的任何分离收缩曲线周围都有一个宽度约为logg2$\frac｛\log g｝｛2｝$的半轴环$X\hbox｛\it；｝$（3）X$X$上最短分离闭合多测地线的长度约为2logg$2\logg$。作为应用，我们还讨论了极值分离收缩期、非简单收缩期的渐近行为，以及当g$g$变为无穷大时最短分离闭合多测地线的预期长度。

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Large genus asymptotics for lengths of separating closed geodesics on random surfaces

In this paper, we investigate basic geometric quantities of a random hyperbolic surface of genus $g$ with respect to the Weil–Petersson measure on the moduli space $M_{g}$ . We show that as $g$ goes to infinity, a generic surface $X \in M_{g}$ satisfies asymptotically:

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