论文标题
大地测量定理和大属的封闭测量学
Prime geodesic theorem and closed geodesics for large genus
论文作者
论文摘要
令$ \ Mathcal {M} _g $为Weil-Petersson Metric赋予的属于$ g $的双曲线表面的模量空间。在本文中，我们表明，对于任何$ε> 0 $，as $ g \ to \ infty $，对于$ \ Mathcal {m} _g $中的通用表面，主要地理定理中的错误术语是由$ g \ g \ cdot t^{\ frac {\ frac {\ frac {3} {3} {3} {4} {4}+subtor从上面限制的。还研究了Prime Geodesic定理中错误术语的预期值。作为一个应用程序，我们表明，作为$ g \ to \ infty $，在$ \ Mathcal {M} _g $最封闭的大地测量学上，长度大大低于$ \ sqrt {g} $是简单且不明显的封闭地位，并且大多数封闭的地源均超过$ \ sqrt { Lipnowski-wright。 当与$ \ sqrt {g+n} $相比，当$ \ Mathcal {m} _ {g，n} $上的相交数字的新颖有效上限也被建立。
Let $\mathcal{M}_g$ be the moduli space of hyperbolic surfaces of genus $g$ endowed with the Weil-Petersson metric. In this paper, we show that for any $ε>0$, as $g\to \infty$, for a generic surface in $\mathcal{M}_g$, the error term in the Prime Geodesic Theorem is bounded from above by $g\cdot t^{\frac{3}{4}+ε}$, up to a uniform constant multiplication. The expected value of the error term in the Prime Geodesic Theorem over $\mathcal{M}_g$ is also studied. As an application, we show that as $g\to \infty$, on a generic hyperbolic surface in $\mathcal{M}_g$ most closed geodesics of length significantly less than $\sqrt{g}$ are simple and non-separating, and most closed geodesics of length significantly greater than $\sqrt{g}$ are not simple, which confirms a conjecture of Lipnowski-Wright. A novel effective upper bound for intersection numbers on $\mathcal{M}_{g,n}$ is also established, when certain indices are large compared to $\sqrt{g+n}$.
