非定向曲面的极小填充对
Minimal Filling pair of non orientable surfaces
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中文总结 AI 辅助
该研究利用胖图理论构造非定向曲面的极小相交填充对，计数映射类群作用下的轨道，证明界随亏格超指数增长，确定最短总长度函数的极小值及相关性质。
中文摘要 AI 辅助
对于g≥3，令N_g表示亏格为g的非定向曲面。本文中，我们通过 fat graphs（胖图）理论构造，建立了N_g上极小相交填充对的存在性。映射类群Mod(N_g)作用于所有此类填充对的集合，我们通过提供下界和上界来计数该作用的Mod(N_g)-轨道。此外，我们利用图上同调证明，这两个界均随g超指数增长。我们还研究了模空间M_g中双曲非定向曲面X上极小相交填充对的长度，定义了函数F_g: M_g→ℝ>0，其中对于X∈M_g，F_g(X)是X上极小相交填充对的最短总长度。我们确定了其最小值m_g，并证明极小值集合与极小相交填充对的Mod(N_g)-轨道一一对应。我们进一步将F_g扩展为Y_g，Y_g定义为在所有填充对上最小化长度，且证明Y_g取得与F_g相同的最小值。
英文摘要
For $g\ge 3$, let $N_g$ denote the non-orientable surface of genus $g$. In this article, we establish the existence of filling pairs on $N_g$ that intersect minimally by construction using the theory of fat graphs. The mapping class group $\mathrm{Mod}(N_g)$ acts on the set of all such filling pairs. We count $\mathrm{Mod}(N_g)$-orbits of this action by providing both lower and upper bounds. Furthermore, we show that both bounds grow super-exponentially with $g$ using graph cohomology. Also, we investigate the lengths of minimally intersecting filling pairs on hyperbolic non-orientable surfaces $X$ in moduli space $\mathcal{M}_g$ of $N_g$. We define a function $\mathcal{F}_g:\mathcal{M}_g\to\mathbb{R}_{>0}$, where for $X\in \mathcal{M}_g$, the function $\mathcal{F}_g(X)$ is the shortest total length of a minimally intersecting filling pair on $X$. We determine its minimum $m_g$ and show that the set of minimizers is in bijection with the \(\mathrm{Mod}(N_g)\)-orbits of minimally intersecting filling pairs. We further extend \(\mathcal{F}_g\) to \(\mathcal{Y}_g\), defined by minimizing the length over all filling pairs, and show that \(\mathcal{Y}_g\) attains the same minimum value as \(\mathcal{F}_g\).
