### Citations

893 | Metric structures for Riemannian and non-Riemannian spaces - Gromov - 1999 |

735 | Three-manifolds with positive Ricci curvature - Hamilton |

510 |
A course in metric geometry
- Burago, Burago, et al.
- 2001
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Citation Context ..., W ′′ 3 , M slim, M2 Section 14.3: W ′2, U ′ 2, W ′′ 2 , M edge, M3 Section 14.4: W ′1, U ′ 1, W ′′ 1 , M 2-stratum Section 15: r∂ , H∂, M ∂ i Appendix B: S, S̃, r(·), W 3. Preliminaries We refer to =-=[BBI01]-=- for basics about length spaces and Alexandrov spaces. 3.1. Pointed Gromov-Hausdorff approximations. Definition 3.1. Let (X, ⋆X) be a pointed metric space. Given δ ∈ [0,∞), two closed subspaces C1 and... |

447 | Ricci flow with surgery on three-manifolds”, arXiv:math/0303109
- Perelman
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Citation Context ...ion. In fact, Theorem 1.3 holds without the second assumption. In order to prove this stronger result, one must use the highly nontrivial Stability Theorem of Perelman [Per91, Kap07]. As mentioned in =-=[Per]-=-, if one does make the second assumption then one can effectively replace the Stability Theorem by standard CK-convergence of Riemannian manifolds. Our proof of Theorem 1.3 is set up so that it extend... |

255 |
Four-manifolds with positive curvature operator
- Hamilton
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Citation Context ...ver the Klein bottle. • 2 ends: S2 × R or T 2 × R. If M has two ends then it splits off an R-factor isometrically. Proof. If M has no end then it is compact and the result follows for C∞-metrics from =-=[Ham86]-=-. For CK-smooth metrics, one could adapt the argument in [Ham86] or alternatively use [Sim09]. 18 BRUCE KLEINER AND JOHN LOTT If M is noncompact then the Cheeger-Gromoll soul theorem says that M is di... |

243 |
AD Alexandrov spaces with curvature bounded below
- Burago, Gromov, et al.
- 1992
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Citation Context ...inate systems which arise from (k, δ)-splittings of Riemannian manifolds, in the presence of a lower curvature bound. The basic construction combines the standard construction of strainer coordinates =-=[BGP92]-=- with the smoothing result of Corollary 3.15. Definition 4.19. Suppose 0 < δ′ ≤ δ, and let α be a (k, δ′)-splitting of a complete pointed Riemannian manifold (M, ⋆M). Let Φ : B ( ⋆M , 1 δ ) → Rk be th... |

185 |
On the structure of complete manifolds of nonnegative curvature
- Cheeger, Gromoll
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Citation Context ...cond assumption in Theorem 1.3.) The topology of nonnegatively curved 3-manifolds is known in the compact case by work of Hamilton [Ham82, Ham86] and in the noncompact case by work of Cheeger-Gromoll =-=[CG72]-=-. In the latter case, the geometry is also well understood. Some relevant examples of such manifolds are: (1) R2 × S1, (2) R× S2, (3) R×Σ, where Σ is a noncompact nonnegatively curved surface which is... |

100 | Notes on Perelman’s papers
- Kleiner, Lott
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Citation Context ...[Per, Theorem 7.4]. Either result can be used to complete the Ricci flow proof of Thurston’s geometrization conjecture. We explain this in Section 17, following the presentation of Perelman’s work in =-=[KL08]-=-. To give a brief explanation, let (M, g(·)) be a Ricci flow with surgery whose initial manifold is compact, orientable and three-dimensional. Put ĝ(t) = g(t) t . Let Mt denote the time t manifold. (... |

89 | Algorithmic topology and classification of 3-manifolds, ACM-Monographs 9 - Matveev - 2003 |

85 |
A Generalized Sphere Theorem
- Grove, Shiohama
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Citation Context ... N ′p has at most one end. The proof of the existence of the scale r0p is based on the fact that nonnegatively curved manifolds are asymptotically conical, the critical point theory of Grove-Shiohama =-=[GS77]-=-, and a compactness argument. Using the approximately conical structure, one obtains a smooth function ηp on 1 r0p M which, when restricted to the metric annulus A(p, 1 10 , 10) ⊂ 1 r0p M , behaves li... |

74 | Almost flat manifolds.
- Gromov
- 1978
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Citation Context ...curvature. Their work characterized the degeneration that can occur when one drops the injectivity radius bound in Gromov’s compactness theorem, generalizing Gromov’s theorem on almost flat manifolds =-=[Gro78]-=-. The corresponding local collapsing structure was used by Anderson and Cheeger-Tian in work on Einstein manifolds [?, CT06]. As far as we know, the first results on collapsing with a lower curvature ... |

60 |
Solution of the Plateau Problem for m-dimensional surfaces of varying topological type
- Reifenberg
- 1960
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Citation Context ...oudy manifolds, and the proof that they have a good manifold core, may be of independent interest. Cloudy manifolds are similar to objects that have been encountered before, in the work of Reifenberg =-=[Rei60]-=- in geometric measure theory and also in [Pug02]. However the clean elementary argument for the existence of a smooth core given in Appendix B, using the universal bundle and transversality, seems to ... |

58 | Critical points of distance functions and applications to geometry. Geometric topology: recent developments (Montecatini Terme, - Cheeger - 1990 |

55 |
On fibering certain 3-manifolds, topology of 3-manifolds and related topics. In:
- Stallings
- 1961
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Citation Context ...is a closed orientable surface then any smooth embedding S → S × R, which is also a homotopy equivalence is isotopic to the fiber S × {0}, as follows from the Schoenflies theorem when S = S2 and from =-=[Sta62]-=- when genus(S) > 0.) (5) follows from the fact that the composition Φ 2 10 , 3 10 , 8 10 , 9 10 ◦ ηp is compactly supported in the annulus A(p, 1 10 , 10). Remark 11.4. One may avoid the Schoenflies... |

48 | Riemannian geometry, volume 171 of Graduate Texts in Mathematics. - Petersen - 2006 |

44 |
Optimization and nonsmooth analysis, volume 5
- Clarke
- 1990
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Citation Context ...g geodesic to m. Take the closed convex hull of the vectors so obtained and then take the intersection as ǫ → 0. This gives a closed convex subset of TmM , which is the generalized gradient of F at m =-=[Cla90]-=-; we will denote this set by ∇genm F . The union⋃ m∈M ∇ gen m F ⊂ TM will be denoted ∇ genF . LOCALLY COLLAPSED 3-MANIFOLDS 19 Lemma 3.13. Let M be a complete Riemannian manifold and let π : TM → M be... |

38 |
Collapsing and pinching under a lower curvature bound,
- Yamaguchi
- 1991
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Citation Context ...lication of the theory of Alexandrov spaces, in particular his Stability Theorem 12 BRUCE KLEINER AND JOHN LOTT from [Per91] (see also [Kap07]); however, these results were never published. Yamaguchi =-=[Yam91]-=- established a fibration theorem for manifolds close to Riemannian manifolds, under a lower curvature bound. Shioya-Yamaguchi [SY00] studied collapsed 3-manifolds with a diameter bound and showed that... |

37 |
Bounding homotopy types by geometry.
- Grove, Petersen
- 1988
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Citation Context ... valid for 2-dimensional Alexandrov spaces. The main difference in the proof of Lemma 9.18 is the method for verifying the fiber topology. For this, we use: Theorem 18.2 (Linear local contractibility =-=[GP88]-=-). For every w ∈ (0,∞) and every positive integer n, there exist r0 ∈ (0,∞) and C ∈ (1,∞) with the following property. If B(p, 1) is a unit ball with compact closure in a Riemannian n-manifold, Rm ∣∣∣... |

37 | Volume collapsed three-manifolds with a lower curvature bound
- Shioya, Yamaguchi
- 2005
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Citation Context ...nly remaining step is to determine the topology of the nonnegatively curved Alexandrov spaces that arise as limits in this fashion. In the case of noncompact limits, this was done by Shioya-Yamaguchi =-=[SY05]-=-. In the compact case, it follows from Simon [Sim09] or, alternatively, from the Ricci flow proof of the elliptization conjecture (using the finite time extinction results of Perelman and Colding-Mini... |

31 | Curvature and injectivity radius estimates for Einstein 4-manifolds, - Cheeger, Tian - 2006 |

30 | Collapsing three-manifolds under a lower curvature bound
- Shioya, Yamaguchi
(Show Context)
Citation Context ...o [Kap07]); however, these results were never published. Yamaguchi [Yam91] established a fibration theorem for manifolds close to Riemannian manifolds, under a lower curvature bound. Shioya-Yamaguchi =-=[SY00]-=- studied collapsed 3-manifolds with a diameter bound and showed that they are graph manifolds, apart from an exceptional case. In [Per], Perelman formulated without proof a theorem equivalent to our T... |

29 | Collapsing Riemannian manifold to ones of lower dimension
- Fukaya
- 1987
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Citation Context ...e used here to be more elegant; moreover, it produces fibrations which are automatically compatible. Embeddings into a Euclidean space were used before to construct fibrations in a collapsing setting =-=[Fuk87]-=-. However, there is the important difference than in the earlier work the base of the fibration was already specified, and this base was embedded into a Euclidean space. In contrast, in the present pa... |

21 | Perelman’s stability theorem
- Kapovitch
(Show Context)
Citation Context ...ure bound were announced by Perelman in the early 90’s, as an application of the theory of Alexandrov spaces, in particular his Stability Theorem 12 BRUCE KLEINER AND JOHN LOTT from [Per91] (see also =-=[Kap07]-=-); however, these results were never published. Yamaguchi [Yam91] established a fibration theorem for manifolds close to Riemannian manifolds, under a lower curvature bound. Shioya-Yamaguchi [SY00] st... |

13 |
Ricci flow of almost non-negatively curved three manifolds
- Simon
(Show Context)
Citation Context ...the nonnegatively curved Alexandrov spaces that arise as limits in this fashion. In the case of noncompact limits, this was done by Shioya-Yamaguchi [SY05]. In the compact case, it follows from Simon =-=[Sim09]-=- or, alternatively, from the Ricci flow proof of the elliptization conjecture (using the finite time extinction results of Perelman and Colding-Minicozzi). For more details, we refer the reader to Sec... |

8 | Collapsing irreducible 3-manifolds with nontrivial fundamental group”, Invent - Bessières, Besson, et al. - 2010 |

8 | A simple proof of Perelman’s collapsing theorem for 3-manifolds”, arXiv:math/1003.2215v3
- Cao, Ge
- 2010
(Show Context)
Citation Context ... collapsing as well as refined results from 3-dimensional topology. Morgan-Tian [MT] gave a proof of Perelman’s collapsing result along the lines of Shioya-Yamaguchi [SY05]. We also mention the paper =-=[CG]-=- by Cao-Ge which relies on more sophisticated Alexandrov space results. 1.6. Acknowledgements. We thank Peter Scott for some references to the 3-manifold literature. 2. Notation and conventions 2.1. P... |

8 | Gromov’s convergence theorem and its application
- Katsuda
- 1985
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Citation Context ...hat these fibrations are compatible on overlaps. To do this, we borrow an idea from the proof of the Whitney embedding theorem (as well as proofs of Gromov’s compactness theorem [Gro99, Chapter 8.D], =-=[Kat85]-=-): we define a smooth map E0 : M → H into a high-dimensional Euclidean space H . The components of E0 are functions of the ηpi’s, the edge function ηE′ , and the scale function p 7→ rp, cutoff appropr... |

8 |
Completion of the Proof of the Geometrization Conjecture”, arXiv:0809.4040
- Morgan, Tian
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Citation Context ...-Porti [BBB+10] gave a different approach to the last part of the proof of the geometrization conjecture, which involves collapsing as well as refined results from 3-dimensional topology. Morgan-Tian =-=[MT]-=- gave a proof of Perelman’s collapsing result along the lines of Shioya-Yamaguchi [SY05]. We also mention the paper [CG] by Cao-Ge which relies on more sophisticated Alexandrov space results. 1.6. Ack... |

6 | Cheeger and Mikhael Gromov. Collapsing Riemannian manifolds while keeping their curvature bounded - Jeff - 1990 |

4 |
Smoothing a Topological Manifold
- Pugh
(Show Context)
Citation Context ...ood manifold core, may be of independent interest. Cloudy manifolds are similar to objects that have been encountered before, in the work of Reifenberg [Rei60] in geometric measure theory and also in =-=[Pug02]-=-. However the clean elementary argument for the existence of a smooth core given in Appendix B, using the universal bundle and transversality, seems to be new. 1.5.9. A sketch of the history. The theo... |