论文标题
大属的封闭双曲线表面的第二个特征值
On second eigenvalues of closed hyperbolic surfaces for large genus
论文作者
论文摘要
在本文中，我们研究了大属的封闭双曲线表面的第二个特征值。我们表明，对于每一个闭合的双曲表面$ x_g $的属$ g $ $（g \ geq 3）$，达到均匀的正常数乘积，第二个eigenvalue $λ_2（x_g）的$ x_g $大于$ x_g $大于$ \ frac { $ \ MATHCAL {L} _2（X_G）$;此外，这两个范围是$ g \ to \ infty $的最佳选择。这里$ \ MATHCAL {L} _2（X_G）$是简单的封闭多晶格的最短长度，将$ x_g $分隔为三个组件。 此外，我们还研究了数量$ \ frac {λ_2（x_g）} {\ mathcal {l} _2（x_g）} $，用于大属的随机双曲线表面。我们表明，作为$ g \ to \ infty $，通用双曲线$ x_g $具有$ \ frac {λ_2（x_g）} {\ Mathcal {l} _2 _2（x_g）} $均匀与$ \ frac {1} {1} {\ ln（g）} $均匀均匀。
In this article, we study the second eigenvalues of closed hyperbolic surfaces for large genus. We show that for every closed hyperbolic surface $X_g$ of genus $g$ $(g\geq 3)$, up to uniform positive constants multiplications, the second eigenvalue $λ_2(X_g)$ of $X_g$ is greater than $\frac{\mathcal{L}_2(X_g)}{g^2}$ and less than $\mathcal{L}_2(X_g)$; moreover these two bounds are optimal as $g\to \infty$. Here $\mathcal{L}_2(X_g)$ is the shortest length of simple closed multi-geodesics separating $X_g$ into three components. Furthermore, we also investigate the quantity $\frac{λ_2(X_g)}{\mathcal{L}_2(X_g)}$ for random hyperbolic surfaces of large genus. We show that as $g\to \infty$, a generic hyperbolic surface $X_g$ has $\frac{λ_2(X_g)}{\mathcal{L}_2(X_g)}$ uniformly comparable to $\frac{1}{\ln(g)}$.
