Structure theory of singular spaces
Structure theory of singular spaces
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DOI：
10.1016/j.jfa.2016.10.020
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发表时间：
2016-03
期刊：
影响因子：
--
通讯作者：
R. Bamler
中科院分区：
文献类型：
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作者：
R. Bamler
在本文中，我们开发了一个结构理论的爱因斯坦流形或流形与较低的里奇曲率界的某些奇异空间，出现的几何极限序列的黎曼流形。该理论推广了Cheeger、Colding和Naber在光滑条件下的结果。在本文的过程中，我们将仔细地描述我们必须对这个黎曼流形序列施加的假设，以保证各个结果保持不变。我们方法的一个重要方面是，我们不需要对导致奇异极限的黎曼流形序列施加任何Ricci曲率界。Ricci曲率界将只需要举行的定期部分的限制，我们不会强加任何（合成）曲率条件的奇异part.The理论在本文中开发的某些几何方程的爆破分析中，我们研究的规模是远远大于当地的曲率规模的应用。特别是，这一理论将有应用的研究里奇流的有界标量曲率，我们将在随后的文件中描述。
In this paper we develop a structure theory of Einstein manifolds or manifolds with lower Ricci curvature bounds for certain singular spaces that arise as geometric limits of sequences of Riemannian manifolds. This theory generalizes the results that were obtained by Cheeger, Colding and Naber in the smooth setting. In the course of the paper, we will carefully characterize the assumptions that we have to impose on this sequence of Riemannian manifolds in order to guarantee that the individual results hold.An important aspect of our approach is that we don't need impose any Ricci curvature bounds on the sequence of Riemannian manifolds leading to the singular limit. The Ricci curvature bounds will only be required to hold on the regular part of the limit and we will not impose any (synthetic) curvature condition on its singular part.The theory developed in this paper will have applications in the blowup analysis of certain geometric equations in which we study scales that are much larger than the local curvature scale. In particular, this theory will have applications in the study of Ricci flows of bounded scalar curvature, which we will describe in a subsequent paper.