On the structure of spaces with Ricci curvature bounded below. II
On the structure of spaces with Ricci curvature bounded below. II
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DOI：
10.4310/jdg/1214342145
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发表时间：
2000
影响因子：
2.5
通讯作者：
J. Cheeger;T. Colding
中科院分区：
文献类型：
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作者：
J. Cheeger;T. Colding
本文和[1]研究了完备连通黎曼流形的列f M i pig的点Gromov-Hausdor极限空间Y的结构，其中的Ricci曲率有一个确定的下界，如RicMn i n，有时也假设有一个体积下界，在这种情况下，称该列f M i pig是不坍缩的。证明了一个收敛序列是非塌缩的当且仅当极限具有正的n维Hausdor测度特别是任何收敛序列要么是塌缩的要么是非塌缩的而且，如果序列是塌缩的，则证明了极限的Hausdor维数实际上是n见第和节我们关于极限空间的无穷小结构的定理在以下方面有等价的陈述：结构上的一个小的，但有限规模的流形与RicMn n铝虽然这两个上下文是重要的，在大多数情况下，它是极限空间，强调在这里通常之间的关系相应的声明流形和极限空间直接遵循的连续性的几何量的问题下，格罗莫夫豪斯多尔限制与格罗莫夫的紧性定理定理也见备注的例子
In this paper and in we study the structure of spaces Y which are pointed Gromov Hausdor limits of sequences f M i pi g of complete connected Riemannian manifolds whose Ricci curvatures have a de nite lower bound say RicMn i n In Sections and sometimes in we also assume a lower volume bound Vol B pi v In this case the sequence is said to be non collapsing If limi Vol B pi then the sequence is said to collapse It turns out that a convergent sequence is noncollapsing if and only if the limit has positive n dimensional Hausdor measure In par ticular any convergent sequence is either collapsing or noncollapsing Moreover if the sequence is collapsing it turns out that the Hausdor dimension of the limit is actually n see Sections and Our theorems on the in nitesimal structure of limit spaces have equivalent statements in terms of or implications for the structure on a small but de nite scale of manifolds with RicMn n Al though both contexts are signi cant for the most part it is the limit spaces which are emphasized here Typically the relation between corre sponding statements for manifolds and limit spaces follows directly from the continuity of the geometric quantities in question under Gromov Hausdor limits together with Gromov s compactness theorem Theorems see also Remark are examples of