论文标题
多边形的模量空间的稳定极限
The Stable Limit of Moduli Spaces of Polygons
论文作者
论文摘要
多边形空间已经进行了广泛的研究，但文献中缺少每个多边形具有的简单属性：维度。这（可能）与多边形所在的环境空间的维度不同。从通常的意义上讲，一个正方形是$ 2 $维的，无论其嵌入环境空间的尺寸如何。如果环境空间的尺寸大于或等于$ 3 $，我们可能会沿对角线弯曲正方形，以产生具有相同边缘长度的$ 3 $维度多边形。然而，即使环境空间的尺寸很大，正方形的弯曲也不会产生大于$ 3 $的多边形。我们概括了这个想法，以表明只有一个具有给定边缘长度的多边形的模量有限，即使环境维度增加而没有结合。
Polygon spaces have been studied extensively, and yet missing from the literature is a simple property that every polygon has: dimension. This is distinct (possibly) from the dimension of the ambient space in which the polygon lives. A square, in the usual sense of the word, is $2$-dimensional no matter the dimension of the ambient space in which it is embedded. If the ambient space has dimension greater than or equal to $3$ we may bend the square along a diagonal to produce a $3$-dimensional polygon with the same edge-lengths. And yet even if the dimension of the ambient space is large, no amount of bending of the square will produce a polygon of dimension larger than $3$. We generalize this idea to show that there are only finitely many moduli spaces of polygons with given edge-lengths, even as the ambient dimension increases without bound.
