The geometry of dented pentagram maps
The geometry of dented pentagram maps
复制标题
凹痕五角星图的几何形状
DOI：
10.14288/1.0043606
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
F. Soloviev
中科院分区：
文献类型：
--
作者：
B. Khesin;F. Soloviev
我们提出了一个新的家庭的自然推广的五角星形地图从二维到高维，并证明其可积性一般扭曲和封闭的多边形。在维$d$有$d-1$这样的推广称为凹痕五角星形地图，我们描述他们的几何形状，连续极限，并与一个谱参数拉克斯表示。我们证明了代数几何可积性的凹痕五角星形地图在3D的情况下，并比较的维数不变环面凹痕地图与那些更高的五角星形地图的帮助下，短对角超平面。当被限制为波纹多边形时，凹陷的五角星形映射在它们自己之间以及与对应的波纹五角星形映射重合。最后，我们证明了可积性的各种五角星形地图的通用和部分波纹多边形在更高的维度。
We propose a new family of natural generalizations of the pentagram map from 2D to higher dimensions and prove their integrability on generic twisted and closed polygons. In dimension $d$ there are $d-1$ such generalizations called dented pentagram maps, and we describe their geometry, continuous limit, and Lax representations with a spectral parameter. We prove algebraic-geometric integrability of the dented pentagram maps in the 3D case and compare the dimensions of invariant tori for the dented maps with those for the higher pentagram maps constructed with the help of short diagonal hyperplanes. When restricted to corrugated polygons, the dented pentagram maps coincide between themselves and with the corresponding corrugated pentagram map. Finally, we prove integrability for a variety of pentagram maps for generic and partially corrugated polygons in higher dimensions.