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## Ricci flow on three-dimensional manifolds with symmetry (2014)

Venue: | Comment. Math. Helv |

Citations: | 3 - 0 self |

### Citations

931 | The entropy formula for the Ricci flow and its geometric applications
- Perelman
- 2002
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Citation Context ...soperimetric inequality. (Cao’s work preceded Perelman’s proof that the product of R and a cigar soliton can never arise as a finite-time dilation limit for the Ricci flow on a compact three-manifold =-=[Per02]-=-.) In the present paper we use new techniques to give general results about the Ricci flow on three-manifolds with symmetries. One of our tools is the result of [Lot10] giving the long-time behavior o... |

501 | The geometries of 3-manifolds - Scott - 1983 |

197 | The formation of singularities in the Ricci flow. - Hamilton - 1995 |

98 | Notes on Perelman’s papers - Kleiner, Lott - 2008 |

76 |
Nilpotent Structures and Invariant Metrics on Collapsed Manifolds
- Cheeger, Fukaya, et al.
- 1992
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Citation Context ...=1 with limi→∞ ci = 0 so that for anym ∈M , the pointed closed metric ball ( B ( m, δ 10 ) , m, gi ) is ci-close to ([− δ 10 , δ 10 ] , 0 ) in the pointed Gromov-Hausdorff topology. Fix δ′ << 1. From =-=[CFG92]-=-, for large i there is a diffeomorphism φi from (M, gi) to (S 1 × S1, g′i), where • g′i is a warped product metric dx2 + f 2i (x)dy2, • maxx∈S1 fi(x)→ 0 as i→∞, and • φi is a eδ′-biLipschitz map. RICC... |

74 | Almost flat manifolds. - Gromov - 1978 |

50 |
Ricci flow of locally homogeneous geometries on closed manifolds
- Isenberg, Jackson
- 1992
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Citation Context ...ows with symmetry have been studied for quite a while. We begin by describing some of the earlier results. Locally homogeneous three-dimensional Ricci flow solutions were examined by Isenberg-Jackson =-=[IJ92]-=- and Knopf-McLeod [KM01]. The flow equations reduced to a system of three coupled ODEs. The solutions are now fairly well understood; see, for example, [Lot07, Section 3]. Certain three-dimensional Ri... |

49 | On the long-time behavior of type-III Ricci flow solutions
- Lott
- 2007
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Citation Context ...uence, we can assume that {(M×[Ai,Ωi], (mi, 0), gi(·), ui(·))}∞i=1 converges to a solution of (2.8) on an étale groupoid. For information about the use of étale groupoids in Ricci flow, we refer to =-=[Lot07]-=- and [Lot10]. The upshot is that after passing to a subsequence, we have smooth pointed convergence to an eternal solution (M∞×R, (m∞, 0), g∞(·), u∞(·)) of (2.8), where M∞ is a two-dimensional étale ... |

41 |
Stability of the Ricci flow at Ricci-flat metrics
- Isenberg, Guenther, et al.
(Show Context)
Citation Context ...trarily small. After passing to a finite cover N̂ , we can assume that there is a universal lower bound on the injectivity radius of the pullback metric ĥ(t). By the linear stability of flat metrics =-=[GIK02]-=-, limt→∞ ĥ(t) exists and is a flat metric. Hence limt→∞ h(t) exists and is a flat metric. Remark 1.5. Theorem 1.1.(iv) says that in the case χ(M) < 0 and as t→∞, over a large part of M the flow (N, h... |

40 | Entropy and reduced distance for Ricci expanders - Feldman, Ilmanen, et al. - 2005 |

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

27 | Evolution of an extended Ricci flow system
- List
- 2005
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Citation Context ...∞ max x∈Xi0 (t) |∇̂u|ĝ(t)(x) = 0. For all sufficiently small i0, if t is sufficiently large then Xi0(t) is nonempty. Remark 1.2. The proof of Theorem 1.1.(i) is essentially contained in List’s paper =-=[Lis08]-=- on a modified Ricci flow. The only thing missing from [Lis08] is the observation that his flow differs from the warped product flow by a Lie derivative. Remark 1.3. The proof of the curvature bound i... |

26 |
Quasi-convergence of model geometries under the Ricci flow
- Knopf, McLeod
(Show Context)
Citation Context ...een studied for quite a while. We begin by describing some of the earlier results. Locally homogeneous three-dimensional Ricci flow solutions were examined by Isenberg-Jackson [IJ92] and Knopf-McLeod =-=[KM01]-=-. The flow equations reduced to a system of three coupled ODEs. The solutions are now fairly well understood; see, for example, [Lot07, Section 3]. Certain three-dimensional Ricci flow solutions with ... |

18 | Dimensional reduction and the long-time behavior of Ricci flow”, Commentarii Mathematici Helvetici
- Lott
- 2010
(Show Context)
Citation Context ...low on a compact three-manifold [Per02].) In the present paper we use new techniques to give general results about the Ricci flow on three-manifolds with symmetries. One of our tools is the result of =-=[Lot10]-=- giving the long-time behavior of a threedimensional Ricci flow solution (N, h(·)) satisfying maxp∈N |RmN |(p, t) = O(t−1) and diam(N, h(t)) = O( √ t). Thus one of our main goals is to show that these... |

15 | The Ricci flow: an introduction, volume 110 - Chow, Knopf - 2004 |

15 | Hamilton’s Ricci flow, volume 77 - Chow, Lu, et al. - 2006 |

13 | Collapsing Riemannian manifolds to ones with lower dimension
- Fukaya
- 1989
(Show Context)
Citation Context ...se, i.e. without an upper diameter bound; see [CT06, Section 2] for discussion. The paper [CFG92] gives a biLipschitz approximation of (M, gi) by a Riemannian nilbundle with affine holonomy; see also =-=[Fuk89]-=-. In the present case such a Riemannian nilbundle is a Riemannian submersion S1 × S1 → S1. For the metric g′i, let Ai be the length of the circle base. We know that Ai ≥ δ10 . To get an upper bound on... |

12 |
Quasi-convergence of Ricci flow for a class of metrics
- Hamilton, Isenberg
- 1993
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Citation Context ... square torus metric [Ham95, Section 11]) it was shown that the Ricci flow exists for all time and converges to a flat metric. Hamilton and Isenberg considered a twisted version of such torus bundles =-=[HI93]-=-. That is, there was a local (with respect to the base) free isometric T 2-action. The T 2-bundle over the circle was globally Date: October 4, 2011. 2010 Mathematics Subject Classification. 53C44,57M... |

10 | Convergence and stability of locally RN -invariant solutions of Ricci - Knopf |

9 |
Quasi-convergence of the Ricci flow
- Knopf
(Show Context)
Citation Context ... Subsection 3.1 for a more precise description. Under the additional assumption of a “solvGowdy” metric, it was shown that the Ricci flow approaches that of a locally homogeneous Sol-metric; see also =-=[Kno00]-=-. Passing to one-dimensional isometry groups, a natural class of geometries comes from warped product metrics with circle fibers and a closed surface base M . One starts with a product metric on N = M... |

6 |
Cheeger and Mikhael Gromov. Collapsing Riemannian manifolds while keeping their curvature bounded
- Jeff
- 1990
(Show Context)
Citation Context ...ature. If the claim were not true then for every ǫ > 0, there would be some tǫ ≥ 1 so that injĝ(tǫ)(m) < ǫ for all m ∈ M . Then M would have an F-structure and hence a vanishing Euler characteristic =-=[CG90]-=-. This is a contradiction. Proposition 2.19. For any i0 > 0, define the i0-thick part of (M, ĝ(t)) by (2.67) Xi0(t) = {m ∈M : injĝ(t)(m) ≥ i0}. Then (2.68) lim t→∞ sup x∈Xi0 (t) |Rĝ(t)(x) + 1| = ... |

3 |
Convergence of the Ricci flow for metrics with indefinite Ricci curvature
- Carfora, Isenberg, et al.
- 1990
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Citation Context ...ns are now fairly well understood; see, for example, [Lot07, Section 3]. Certain three-dimensional Ricci flow solutions with a two-dimensional isometry group were analyzed by Carfora-Isenberg-Jackson =-=[CIJ90]-=- and Hamilton [Ham95, Section 11]. They considered Riemannian threemanifolds that admit a free isometric T 2-action. The base is a circle and the total space is (necessarily) diffeomorphic to a 3-toru... |

2 | Collapsed manifolds with bounded sectional curvature and applications - Rong - 2007 |

1 |
Isoperimetric estimate for the Ricci flow on S2
- Cao
- 2005
(Show Context)
Citation Context ...K04, Chapter 5] break down. When M is a two-sphere, Xiaodong Cao showed that the product of R and a cigar soliton cannot arise as a finite-time dilation limit of a warped product Ricci flow on S2×S1. =-=[Cao05]-=-. His argument used an isoperimetric inequality. (Cao’s work preceded Perelman’s proof that the product of R and a cigar soliton can never arise as a finite-time dilation limit for the Ricci flow on a... |