收缩Ricci孤立子与Ricci流的epsilon-正则性

Cheeger and Tian (J Am Math Soc 19(2):487–525, 2006) proved an ϵ -regularity theorem for 4-dimensional Einstein manifolds without volume assumption. They conjectured that similar results should hold for critical metrics with constant scalar curvature, shrinking Ricci solitons, Ricci flows in 4-dimensional manifolds and higher dimensional Einstein manifolds. In this paper we consider all these problems. First, we construct counterexamples to the conjecture for 4-dimensional critical metrics and counterexamples to the conjecture for higher dimensional Einstein manifolds. For 4-dimensional shrinking Ricci solitons, we prove an ϵϵ -regularity theorem which confirms Cheeger–Tian’s conjecture with a universal constant ϵϵ . For Ricci flow, we reduce Cheeger–Tian’s ϵϵ -regularity conjecture to a backward Pseudolocality estimate. By proving a global backward Pseudolocality theorem, we can prove a global ϵϵ -regularity theorem which partially confirms Cheeger–Tian’s conjecture for Ricci flow. Furthermore, as a consequence of the ϵϵ -regularity, we can show by using the structure theorem of Naber and Tian (Geometric structures of collapsing Riemannian manifolds I. Surveys in geometric analysis and relativity, International Press, Somerville, 2011) that a collapsed limit of shrinking Ricci solitons with bounded L2 curvature has a smooth Riemannian orbifold structure away from a finite number of points

切格（Cheeger）和田刚（Tian）（《美国数学会杂志》19(2):487 - 525, 2006）在无体积假设的情况下证明了一个关于4维爱因斯坦流形的\(\epsilon\)-正则性定理。他们推测类似的结果对于具有常数量曲率的临界度量、收缩里奇孤立子、4维流形中的里奇流以及高维爱因斯坦流形应该成立。在本文中我们考虑所有这些问题。首先，我们构造了针对4维临界度量的猜想的反例以及针对高维爱因斯坦流形的猜想的反例。对于4维收缩里奇孤立子，我们证明了一个\(\epsilon\)-正则性定理，该定理以一个通用常数\(\epsilon\)证实了切格 - 田刚的猜想。对于里奇流，我们将切格 - 田刚的\(\epsilon\)-正则性猜想归结为一个反向伪局部性估计。通过证明一个全局反向伪局部性定理，我们能够证明一个全局\(\epsilon\)-正则性定理，该定理部分地证实了切格 - 田刚关于里奇流的猜想。此外，作为\(\epsilon\)-正则性的一个结果，我们可以利用纳伯（Naber）和田刚的结构定理（《坍缩黎曼流形的几何结构I. 几何分析与相对论综述》，国际出版社，萨默维尔，2011）表明，具有有界\(L^2\)曲率的收缩里奇孤立子的坍缩极限在除去有限个点之外具有一个光滑的黎曼轨形结构。