Manifolds with positive curvature operators are space forms
Manifolds with positive curvature operators are space forms
复制标题
DOI：
10.4007/annals.2008.167.1079
复制
发表时间：
2006-06
影响因子：
4.9
通讯作者：
Christoph E. Boehm;Burkhard Wilking
中科院分区：
文献类型：
--
作者：
Christoph E. Boehm;Burkhard Wilking
里奇流由 Hamilton 于 1982 年引入 [H1]，以证明承认正里奇曲率的黎曼度量的紧致三流形是球空间形式。在第四维中，汉密尔顿证明具有正曲率算子的紧凑四流形也是球面空间形式[H2]。更一般地说，同样的结论适用于具有 2-正曲率算子的紧凑四流形 [Che]。回想一下，如果曲率算子的两个最小特征值之和为正，则该算子称为 2-正。在任意维度中，Huisken [Hu] 描述了曲率算子空间中的显式开锥体，使得归一化 Ricci 流将其曲率算子包含在该锥体中的度量演化为恒定正截面曲率的度量。汉密尔顿猜想，在所有维度上，具有正曲率算子的紧黎曼流形必定是空间形式。在本文中我们证实了这个猜想。更一般地，我们展示以下内容
The Ricci flow was introduced by Hamilton in 1982 [H1] in order to prove that a compact three-manifold admitting a Riemannian metric of positive Ricci curvature is a spherical space form. In dimension four Hamilton showed that compact four-manifolds with positive curvature operators are spherical space forms as well [H2]. More generally, the same conclusion holds for compact four-manifolds with 2-positive curvature operators [Che]. Recall that a curvature operator is called 2-positive, if the sum of its two smallest eigenvalues is positive. In arbitrary dimensions Huisken [Hu] described an explicit open cone in the space of curvature operators such that the normalized Ricci flow evolves metrics whose curvature operators are contained in that cone into metrics of constant positive sectional curvature. Hamilton conjectured that in all dimensions compact Riemannian manifolds with positive curvature operators must be space forms. In this paper we confirm this conjecture. More generally, we show the following