Collapsing and pinching under a lower curvature bound
Collapsing and pinching under a lower curvature bound
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DOI：
10.2307/2946563
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发表时间：
1991-03
影响因子：
4.9
通讯作者：
Takao Yamaguchi
中科院分区：
文献类型：
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作者：
Takao Yamaguchi
本文研究了截面曲率一致有界的黎曼流形的坍缩现象和拼挤问题。对于正整数n和D > 0，设X/是具有截面曲率KM的紧致黎曼n-流形M的集合？-1和直径diam（M）< D。根据Gromov [GLP]的弱紧性定理，X'在紧度量空间集合中相对于Hausdorff距离是相对紧的. Gromov在[G2]中进一步证明了X中任意元素的Betti数之和关于给定的常数是一致有界的。这些结果表明，X/是一个研究对象，我们可以开发一些几何和拓扑。事实上，当体积一致有界远离零时，下面的结果是已知的。令#（v）是具有体积vol（M）的X/的子集？v.格罗夫和Petersen [GP 1]证明了4（v）中元素的同伦类型集是有限的。（相关结果见[Y2]。）对于Pinching问题，大津，Shiohama和作者（[OSY]，[Y3]）在X（v）类中得到了一些微分球定理.根据最近的一篇文章[GPW]，如果n #3，4，则（v）至多包含1000个复同态类型.设Mi（i = 1，2，.）是A中的收敛序列，X是它们的极限。如果X的Hausdorff维数小于n，我们说Mi坍缩到X。在体积条件下，4（v）中没有发生塌缩。从这个观点出发，我们很自然地要问，在类X/中发生了什么样的坍缩现象，什么样的拼挤定理成立。在这种情况下，似乎很难确定X的奇点（第1节给出了一些例子）。本文研究了X是黎曼流形时的坍缩现象，建立了一个拼挤定理。对于黎曼流形M，我们用inj（M）表示M的内射性半径。当N被写在X的位置并且N的度量被归一化时，我们的
In this paper we are concerned with collapsing phenomena and pinching problems of Riemannian manifolds whose sectional curvatures are uniformly bounded from below. For a positive integer n and for D > 0, let X/ be the set of compact Riemannian n-manifolds M with sectional curvatures KM ? -1 and diameters diam(M) < D. By the weak compactness theorem of Gromov [GLP], X' is relatively compact in the set of compact metric spaces with respect to the Hausdorff distance. Furthermore Gromov proved in [G2] that the sum of Betti numbers of any element in X is uniformly bounded in terms of the given constants. These results suggest that X/ is an object of study on which we could develop some geometry and topology. In fact, when volumes are uniformly bounded away from zero, the following results are known. Let #(v) be the subset of X/ with volume vol(M) ? v. Grove and Petersen [GP1] proved that the set of homotopy types of elements in 4(v) is finite. (Related results are in [Y2].) For pinching problems Otsu, Shiohama and the author ([OSY], [Y3]) obtained some differential sphere theorems in the class X(v). According to a recent paper [GPW], if n # 3,4, then (v) contains at most finitely many diffeomorphism types. Let Mi (i = 1, 2,...) be a convergent sequence in A, and X be the limit of them. We say that Mi collapses to X if the Hausdorff dimension of X is less than n. By the volume condition, no collapsing occurs in 4(v). From this point of view it is quite natural to ask what collapsing phenomena occur and what types of pinching theorems hold in the class X/. In this situation it seems difficult to determine the singularities of X (some examples are given in Section 1). In this paper we study collapsing phenomena in the case when X is a Riemannian manifold and establish a pinching theorem. For a Riemannian manifold M we denote by inj(M) the injectivity radius of M. When N is written in place of X and the metric of N is normalized, our