Isoparametric submanifolds and their Coxeter groups
Isoparametric submanifolds and their Coxeter groups
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DOI：
10.4310/jdg/1214439466
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发表时间：
1985
影响因子：
2.5
通讯作者：
C. Terng
中科院分区：
文献类型：
--
作者：
C. Terng
0. 引言 1930 年代后期，Elie Cartan 在空间形式 TV 上定义了等参函数的概念，并开始了他们的研究 [4]-[7]。平滑函数/：N -> R (N = ^ " + 1 , S 或 H\ 是等参函数，如果 Δ/ 和 | v/ | 2 是 / 的函数。除其他外，嘉当证明 / 的水平超曲面是平行的，并且每个超曲面都有恒定的主曲率。反之，他表明，如果 M 是具有恒定主曲率的 N 的超曲面，则至少存在一个以 M 作为水平的局部等参函数。嘉当称这样的为在过去的十年里，许多人推进了这项研究[19, 25]，最终在 1980 年左右，Mύnzner [18] 完成了球体中等参超曲面的美丽结构理论，从而将其分类简化为一个（困难！）代数问题。随后许多人对这个分类问题做出了贡献，包括 U. Abresch [1]、D. Ferus、H. Karcher、H. F. Mύnzner [15] 等人虽然最近取得了相当大的进展，但总的来说，等参超曲面理论本身就是一个特殊的主题；然而近年来，调和映射理论 [12] 和最小子流形 [14,19,23] 得到了应用。最近，Eells [12] 给出了用于构建调和映射的等参映射的定义。 S. Carter 和 A. West [3] 给出了从 N 到 R 的等参映射的更强的定义；他们的目的是将 Cartan 的工作推广到更高的余维，他们能够证明存在与每个等参映射/：N -> R 相关的 Coxeter 群（即，由反射生成的有限群）。但是，他们也无法为较大的 m 构造一个全局等参映射。给定的等参子流形。
0. Introduction In the later 1930's Elie Cartan defined the notion of isoparametric functions on a space form TV and began their study [4]-[7]. A smooth function/: N -> R (N = ^ " + 1 , S or H\ is isoparametric, if Δ/and | v/ | 2 are functions of /. Among other things Cartan showed that the level hypersurfaces of / are parallel, and each has constant principal curvatures. And conversely, he showed that if M is a hypersurface of N with constant principal curvatures, then there is at least a local isoparametric function having M as a level. Cartan called such a hypersurface isoparametric. In the last ten years, many people carried forward this research [19, 25]. Finally around 1980, Mύnzner [18] completed the beautiful structure theory of isoparametric hypersurfaces in the spheres, and thereby reduced their classification to a (difficult!) algebraic problem. Many people subsequently made contributions to this classification problem including U. Abresch [1], D. Ferus, H. Karcher, H. F. Mύnzner [15], et al. While there has been considerable recent progress, it seems much remains to be done. By and large, the theory of isoparametric hypersurfaces has been a special subject by itself; however in recent years there have been applications to the theory of harmonic maps [12], and minimal submanifolds [14, 19, 23]. Recently, Eells [12] gave a definition of isoparametric maps for the purpose of constructing harmonic maps. S. Carter and A. West [3] gave a stronger definition of isoparametric maps from N to R; their purpose being to generalize Cartan's work to higher codimension. Using their definitions, they were able to show that there is a Coxeter group (i.e., a finite group generated by reflections) associated to each isoparametric map/: N -> R. However, they did not obtain a similar result for larger m. They were also unable to construct a global isoparametric map for a given isoparametric submanifold.