A Simple Proof of Perelman’s Collapsing Theorem for 3-manifolds

@article{Cao2010ASP,
  title={A Simple Proof of Perelman’s Collapsing Theorem for 3-manifolds},
  author={Jianguo Cao and Jianquan Ge},
  journal={Journal of Geometric Analysis},
  year={2010},
  volume={21},
  pages={807-869}
}
We will simplify earlier proofs of Perelman’s collapsing theorem for 3-manifolds given by Shioya–Yamaguchi (J. Differ. Geom. 56:1–66, 2000; Math. Ann. 333: 131–155, 2005) and Morgan–Tian (arXiv:0809.4040v1 [math.DG], 2008). A version of Perelman’s collapsing theorem states: “Let$\{M^{3}_{i}\}$be a sequence of compact Riemannian 3-manifolds with curvature bounded from below by (−1) and$\mathrm{diam}(M^{3}_{i})\ge c_{0}>0$. Suppose that all unit metric balls in$M^{3}_{i}$have very small volume… 

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