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## ON COMPLETE GRADIENT SHRINKING RICCI SOLITONS (2009)

Citations: | 55 - 6 self |

### Citations

938 | The entropy formula for the Ricci flow and its geometric applications, eprint - Perelman |

901 | Einstein manifolds - Besse - 1986 |

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

447 | Ricci flow with surgery on three-manifolds - Perelman |

197 |
The formation of singularities in the Ricci flow.
- Hamilton
- 1995
(Show Context)
Citation Context ...ent shrinking Ricci solitons and prove Theorem 1.1. First of all, we need a few useful facts about complete gradient shrinking solitons. The first basic result is due to Hamilton (cf. Theorem 20.1 in =-=[9]-=-). Lemma 2.1. Let (M n , gij, f) be a complete gradient shrinking soliton satisfying (1.1). Then we have ∇iR = 2Rij∇jf, and R + |∇f| 2 − f = C0 for some constant C0. Here R denotes the scalar curvatur... |

163 |
The formation of singularities in the Ricci flow, Surveys in differential geometry,
- Hamilton
- 1993
(Show Context)
Citation Context ...ent shrinking Ricci solitons and prove Theorem 1.1. First of all, we need a few useful facts about complete gradient shrinking solitons. The first basic result is due to Hamilton (cf. Theorem 20.1 in =-=[9]-=-). Lemma 2.1. Let (M n , gij, f) be a complete gradient shrinking soliton satisfying (1.1). Then we have ∇iR = 2Rij∇jf, and R + |∇f| 2 − f = C0 for some constant C0. Here R denotes the scalar curvatur... |

112 |
A complete proof of the Poincare and geometrization conjecturesapplication of the Hamilton-Perelman theory of the Ricci flow,
- Cao, Zhu
- 2006
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Citation Context ...l Bx0(1). This completes the proof of Proposition 2.1 and Theorem 1.1. □ 2 Indeed, the above argument was essentially sketched by Perelman (see, p.3 of [13]), and a detailed argument was presented in =-=[4]-=- (p.385-386).6 HUAI-DONG CAO AND DETANG ZHOU 3. Volume growth of complete gradient shrinking solitons In this section, we examine the volume growth of geodesic balls of complete noncompact gradient s... |

92 | Strong uniqueness of the Ricci flow.
- Chen
- 2009
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Citation Context ...will make this normalization on f throughout the paper. We will also need the following useful result, which is a special case of a more general result on complete ancient solutions due to B.-L. Chen =-=[6]-=- (cf. Proposition 5.5 in [1]). Lemma 2.2. Let (M n , gij, f) be a complete shrinking Ricci soliton. Then gij has nonnegative scalar curvature R ≥ 0. As an immediate consequence of (2.1) and Lemma 2.2,... |

92 |
Rotationally symmetric shrinking and expanding gradient Kahler-Ricci solitons.
- Feldman, Ilmanen, et al.
- 2003
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Citation Context ...(x) ≤ 1 4 (r(x) + c)2 holds for any complete noncompact gradient shrinking soliton. It remains interesting to find out whether R is bounded from above by a constant. Remark 1.4. Feldman-Ilmanen-Knopf =-=[8]-=- constructed a complete noncompact gradient Kähler shrinker on the tautological line bundle O(−1) of the complex projective space CP n−1 (n ≥ 2) which has Euclidean volume growth, quadratic curvature ... |

77 | Comparison geometry for the Bakry-Emery Ricci tensor,
- Wei, Wylie
- 2009
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Citation Context ... ≥ 0 must have zero asymptotic volume ratio, i.e., limr→∞ Vol(Bx0(r))/r n = 0. Combining Theorem 1.1 and Theorem 1.2, we also have the following consequence, which was obtained previously in [10] and =-=[15]-=- respectively. Corollary 1.1. Let (Mn, gij, f) be a complete noncompact gradient shrinking Ricci soliton. Then we have ∫ |u|e −f dV < +∞ for any function u on M with |u(x)| ≤ Aeαr2 (x) 1 , 0 ≤ α < 4 p... |

62 |
Manifolds with density
- Morgan
(Show Context)
Citation Context ...vature Rc ≥ 0 must have zero asymptotic volume ratio, i.e., limr→∞ Vol(Bx0(r))/r n = 0. Combining Theorem 1.1 and Theorem 1.2, we also have the following consequence, which was obtained previously in =-=[10]-=- and [15] respectively. Corollary 1.1. Let (Mn, gij, f) be a complete noncompact gradient shrinking Ricci soliton. Then we have ∫ |u|e −f dV < +∞ for any function u on M with |u(x)| ≤ Aeαr2 (x) 1 , 0 ... |

46 | Zur Weylschen Relativitatstheorie und der Weylschen Erweiterung des Krummungsbegriffs. - Bach - 1921 |

42 | On locally conformally flat gradient steady Ricci solitons - Cao, Chen |

42 | Remarks on non-compact gradient Ricci solitons - Pigola, Rimoldi, et al. - 2011 |

35 | On the classification of gradient Ricci solitons - Petersen, Wylie |

34 | On a classification of gradient shrinking solitons
- Ni, Wallach
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Citation Context ...g0) with the potential function |x| 2 /4, the leading term 1 4r2 (x) for the lower and upper bounds on f in Theorem 1.1 is optimal. We also point out that it has been known, by the work of Ni-Wallach =-=[12]-=- and Cao-Chen-Zhu [3], that any 3dimensional complete noncompact non-flat shrinking gradient soliton is necessarily the round cylinder S2 × R or one of its Z2 quotients. 1 The first author was partial... |

34 | Recent progress on Ricci solitons, in Recent Advances in Geometric Analysis, - Cao - 2009 |

31 | On gradient Ricci solitons - Munteanu, Sesum |

27 | Sharp logarithmic Sobolev inequalities on gradient solitons and applications
- Carrillo, Ni
(Show Context)
Citation Context ...f the form f(x) ≥ 1 8 r2 (x) − c ′ 1 was shown by Ni [11]. Moreover, the upper bound in Theorem 1.1 was essentially observed in [3], while a rough quadratic lower bound, as pointed out by Carrillo-Ni =-=[5]-=-, could follow from the argument of Fang-Man-Zhang in [7]. Theorem 1.2. Let (M n , gij, f) be a complete noncompact gradient shrinking Ricci soliton. Then, there exists some positive constant C1 > 0 s... |

26 | Recent developments on Hamilton’s Ricci flow, Surveys in differential geometry
- Cao, Chen, et al.
- 2008
(Show Context)
Citation Context ... function |x| 2 /4, the leading term 1 4r2 (x) for the lower and upper bounds on f in Theorem 1.1 is optimal. We also point out that it has been known, by the work of Ni-Wallach [12] and Cao-Chen-Zhu =-=[3]-=-, that any 3dimensional complete noncompact non-flat shrinking gradient soliton is necessarily the round cylinder S2 × R or one of its Z2 quotients. 1 The first author was partially supported by NSF g... |

26 | Gradient shrinking solitons with vanishing Weyl tensor - Zhang |

22 | On locally conformally flat gradient shrinking Ricci solitons, preprint - Cao, Wang - 2008 |

19 | Geometry of complete gradient shrinking Ricci solitons, Geometric and Analysis (Vol I - Cao - 2011 |

17 | Complete gradient shrinking Ricci solitons have finite topological type,
- Fang, Man, et al.
- 2008
(Show Context)
Citation Context .... Moreover, the upper bound in Theorem 1.1 was essentially observed in [3], while a rough quadratic lower bound, as pointed out by Carrillo-Ni [5], could follow from the argument of Fang-Man-Zhang in =-=[7]-=-. Theorem 1.2. Let (M n , gij, f) be a complete noncompact gradient shrinking Ricci soliton. Then, there exists some positive constant C1 > 0 such that for r > 0 sufficiently large. Vol(Bx0(r)) ≤ C1r ... |

17 |
Ancient solutions to kähler-ricci
- Ni
- 2005
(Show Context)
Citation Context ...vature of (M n , gij, f) is assumed to be bounded, Theorem 1.1 was shown by Perelman [13]. Also, under the assumption of Rc ≥ 0, a lower estimate of the form f(x) ≥ 1 8 r2 (x) − c ′ 1 was shown by Ni =-=[11]-=-. Moreover, the upper bound in Theorem 1.1 was essentially observed in [3], while a rough quadratic lower bound, as pointed out by Carrillo-Ni [5], could follow from the argument of Fang-Man-Zhang in ... |

16 | Bach-flat gradient steady Ricci solitons - Cao, Catino, et al. |

16 | On four-dimensional anti-self-dual gradient Ricci solitons - Chen, Wang - 2011 |

13 | Rigidity of shrinking Ricci solitons - Fernández-López, Garćıa-Ŕıo |

10 | Uniqueness of gradient Ricci solitons - Brendle |

10 | On rotationally invariant shrinking Ricci solitons - Kotschwar |

9 |
On complete gradient shrinking solitons
- Cao, Zhou
(Show Context)
Citation Context ...(3.9) V ′ ∫ r (s) ds ≥ V (s) 1 V (r) ≥ V (1)r n−2δ . n − 2δ ds. s Vol(Bx0(r)) ≥ V (r − c) ≥ 2 −n V (1)r n−2δ for r sufficiently large. □ Remark 3.2. X.-P. Zhu and the first author (see Theorem 3.1 in =-=[2]-=-) have shown that a complete noncompact gradient shrinking soliton, without any curvature assumption, must have infinite volume. Their proof is, however, more sophisticated, relying on a logarithmic i... |

8 |
G.,Ricci flow with Surgery on Three-manifolds
- Perelmann
- 2003
(Show Context)
Citation Context ...as partially supported by CNPq and FAPERJ, Brazil. 12 HUAI-DONG CAO AND DETANG ZHOU Remark 1.2. When the Ricci curvature of (M n , gij, f) is assumed to be bounded, Theorem 1.1 was shown by Perelman =-=[13]-=-. Also, under the assumption of Rc ≥ 0, a lower estimate of the form f(x) ≥ 1 8 r2 (x) − c ′ 1 was shown by Ni [11]. Moreover, the upper bound in Theorem 1.1 was essentially observed in [3], while a r... |

5 | Evolution of the Weyl tensor under the Ricci - Catino, Mantegazza |

4 |
Recent progress on Ricci solitons ,to appear
- Cao
(Show Context)
Citation Context ... on f throughout the paper. We will also need the following useful result, which is a special case of a more general result on complete ancient solutions due to B.-L. Chen [6] (cf. Proposition 5.5 in =-=[1]-=-). Lemma 2.2. Let (M n , gij, f) be a complete shrinking Ricci soliton. Then gij has nonnegative scalar curvature R ≥ 0. As an immediate consequence of (2.1) and Lemma 2.2, one gets the following resu... |

4 | Ricci solitons: the equation point of view - unknown authors |

1 | Self-dual K ahler manifolds and Einstein manifolds of dimension four - Derdzinski - 1983 |