A pair $(\alpha, \beta)$ of simple closed geodesics on a closed and oriented hyperbolic surface $M_g$ of genus $g$ is called a filling pair if the complementary components of $\alpha\cup\beta$ in $M_g$ are simply connected. The length of a filling pair is defined to be the sum of their individual lengths. We show that the length of any filling pair on $M$ is at least $\frac{m_{g}}{2}$, where $m_{g}$ is the perimeter of the regular right-angled hyperbolic $\left(8g-4\right)$-gon.

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关于填充对长度的猜想

如果 $\alpha\cup\beta$ 在 $\alpha\cup\beta$ 中的互补分量在 $g$ 属的闭合和定向双曲表面 $M_g$ 上的一对 $(\alpha,\beta)$ 简单闭合测地线称为填充对M_g$ 是简单连接的。填充对的长度定义为它们各自长度的总和。我们证明 $M$ 上任何填充对的长度至少为 $\frac{m_{g}}{2}$，其中 $m_{g}$ 是规则直角双曲线 $\left 的周长(8g-4\right)$-gon。