Department of Mathematics, University of California, Berkeley, California 94720.
Proc Natl Acad Sci U S A. 1982 Jun;79(12):3931-2. doi: 10.1073/pnas.79.12.3931.
Mean curvature is one of the simplest and most basic of local differential geometric invariants. Therefore, closed hypersurfaces of constant mean curvature in euclidean spaces of high dimension are basic objects of fundamental importance in global differential geometry. Before the examples of this paper, the only known example was the obvious one of the round sphere. Indeed, the theorems of H. Hopf (for immersion of S(2) into E(3)) and A. D. Alexandrov (for imbedded hypersurfaces of E(n)) have gone a long way toward characterizing the round sphere as the only example of a closed hypersurface of constant mean curvature with some added assumptions. Examples of this paper seem surprising and are constructed in the framework of equivariant differential geometry.
平均曲率是最简单、最基本的局部微分几何不变量之一。因此，高维欧几里得空间中具有常平均曲率的闭超曲面是整体微分几何中具有基本重要性的基本对象。在本文的例子之前，唯一已知的例子是明显的圆球面。事实上，H. Hopf（关于 S(2)浸入 E(3)）和 A. D. Alexandrov（关于 E(n)中的嵌入超曲面）的定理已经在很大程度上把具有某些附加假设的常平均曲率闭超曲面的唯一例子刻画为圆球面。本文中的例子似乎令人惊讶，并且是在等变微分几何的框架下构造的。
