Volume collapsed three-manifolds with a lower curvature bound

By T. Shioya, Takao Yamaguchi. 2003

79 citations recorded by the source corpus.

In this paper we determine the topology of three-dimensional complete orientable Riemannian manifolds with a uniform lower bound of sectional curvature whose volume is sufficiently small.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e83fc8b4955f12cb5fba750a8e541edeb240ba91

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In Applied Mathematics

1. Xiping Zhu
2. J. Lott
3. Huaihuo Cao
4. Bruce Kleiner
5. G. Tian
6. Binglong Chen
7. V. Kapovitch
8. Nicola Gigli

In Geometry and Topology

1. J. Lott
2. Xiping Zhu
3. Huaihuo Cao
4. Bruce Kleiner
5. Binglong Chen
6. G. Tian
7. V. Kapovitch
8. J. Morgan

Citations in the corpus

• Dimensional reduction and the long-time behavior of Ricci flow (2007, background): “Perelman showed that the nonhyperbolic part of the evolving manifold satisfies certain geometric conditions, from which one can show that it is a graph manifold [1, 32, 42, 44, 51].”
• Hamilton-Perelman's Proof of the Poincar\'e Conjecture and the Geometrization Conjecture (2006, background): “In the last section of this paper (see Theorem 7.7.1), we will provide a proof of Thurston’s geometrization conjecture by only using Shioya-Yamaguchi’s collapsing result.”
• Notes on Perelman's papers (2006, methodology): “[52, 64] Let Yt be the truncation of ⋃k i=1Hi obtained by removing horoballs at distance approximately 1 2α(t) from the basepoints xi.”
• Perelman???s Stability Theorem (2007, background): “However, as was communicated to the author by Kleiner, for that particular application, if one traces through the proofs of [26, 27] carrying along the additional bounds arising from the Ricci flow, then one finds that in each instance when a 3-dimensional Alexandrov space arises as a Gromov-Hausdor”
• Recent progress on the Poincaré conjecture and the classification of 3-manifolds (2004, background): “For results on collapsing metrics with lower curvature bound, see for example [22] and [23].”
• Completion of the Proof of the Geometrization Conjecture (2008, background): “ng to the Ricci flow results from [18] and the material in [19] preceding Theorem 7.4). A version of Theorem 7.4 for closed 3-manifolds has been proved in a series of papers of Shioya-Yamaguchi ([22], [23]). They did not make use of Assumption 3 on bounds on derivatives of curvature.5 So their result”
• Recent developments on the Hamilton’s Ricci Flow (2007, unknown): “For case (i), we will appeal to the following collapsing result, which was first announced (in a more general version) by Perelman [81] and proved by Shioya-Yamaguchi [ 92 ].”
• A Complete Proof of the Poincaré and Geometrization Conjectures - application of the Hamilton-Perelman theory of the Ricci flow (2006, background): “In the last section of this paper (see Theorem 7.7.1), we will provide a proof of Thurston’s geometrization conjecture by only using Shioya-Yamaguchi’s collapsing result.”