Minimally intersecting filling pairs on surfaces
Minimally intersecting filling pairs on surfaces
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表面上最小相交的填充对
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
ShinnYih Huang
中科院分区：
文献类型：
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作者：
Tarik Aougab;ShinnYih Huang
设Sg表示g属的闭合可定向曲面。通过证明这些轨道与对称群中某置换方程的解相对应，构造了由填充Sg且相交最小的简单闭曲线对构成的指数多映射类群轨道。接下来，我们证明了最小相交填充对是组合最优的，因为有许多简单的封闭曲线与填充对正好相交一次。最后，我们开始了g属黎曼曲面模空间上的拓扑莫尔斯函数Fg的研究，给出了一个双曲度规，输出了该度规的最短最小相交填充对的长度。我们完整地刻画了Fg的全局极小值，并利用本文第一部分构造的最小相交填充对的指数多映射类群轨道，证明了这种极小值的数量在g中至少呈指数增长。57 m20, 57 m50
Let Sg denote the closed orientable surface of genus g . We construct exponentially many mapping class group orbits of pairs of simple closed curves which fill Sg and intersect minimally, by showing that such orbits are in correspondence with the solutions of a certain permutation equation in the symmetric group. Next, we demonstrate that minimally intersecting filling pairs are combinatorially optimal, in the sense that there are many simple closed curves intersecting the pair exactly once. We conclude by initiating the study of a topological Morse function Fg over the moduli space of Riemann surfaces of genus g , which, given a hyperbolic metric , outputs the length of the shortest minimally intersecting filling pair for the metric . We completely characterize the global minima of Fg and, using the exponentially many mapping class group orbits of minimally intersecting filling pairs that we construct in the first portion of the paper, we show that the number of such minima grows at least exponentially in g . 57M20, 57M50