Reconstruction of Collapsed Manifolds
Reconstruction of Collapsed Manifolds
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DOI：
10.1142/9789814324359_0079
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发表时间：
2011-06
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通讯作者：
Takao Yamaguchi
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文献类型：
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作者：
Takao Yamaguchi
在这篇文章中，我们考虑了利用极限空间的几何或解析数据在模空间中重建折叠流形的问题。我们主要感兴趣的模空间是由固定维闭黎曼流形组成的，它具有较低的截面曲率和较高的直径界。在这个模空间中，我们可以根据极限Alexandrov空间的奇点重构三维或四维折叠流形的拓扑。在一般维上，我们定义了一个新的覆盖不变量，并利用Gromov的Betti数定理证明了它的一致有界性。最后，我们利用解析谱数据讨论了折叠流形的重构和稳定性问题，其中我们假设了一个附加的截面曲率上限。
In this article, we consider the problem of reconstructing collapsed manifolds in a moduli space by means of geometric or analytic data of the limit spaces. The moduli space of our main interest is that consisting of closed Riemannian manifolds of fixed dimension with a lower sectional curvature and an upper diameter bound. In this moduli space, we can reconstruct the topology of threedimensional or four-dimensional collapsed manifolds in terms of the singularities of the limit Alexandrov spaces. In the general dimension, we define a new covering invariant and prove the uniform boundedness of it with an application to Gromov’s Betti number theorem. Finally we discuss the reconstruction and stability problems of collapsed manifolds by using analytic spectral data, where we assume an additional upper sectional curvature bound.