Ricci Flow with Surgery on Four-manifolds with Positive Isotropic Curvature

By Binglong Chen, Xiping Zhu. 2005

100 citations recorded by the source corpus.

In this paper we study the Ricci flow on compact four-manifolds with positive isotropic curvature and with no essential incompressible space form. Our purpose is two-fold. One is to give a complete proof of Hamilton's classification theorem on four-manifolds with positive isotropic curvature and with no essential incompressible space form; the other is to extend some recent results of Perelman on the three-dimensional Ricci flow to four-manifolds. During the the proof we have actually provided, up to slight modifications, all necessary details for the part from Section 1 to Section 5 of Perelman's second paper on the Ricci flow.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/dd76f595285fce21817d68e86393aa4c8c735860

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Huaihuo Cao
2. Binglong Chen
3. Xiping Zhu
4. S. Brendle
5. Peter Petersen
6. J. Lott
7. Xiaohua Zhu
8. Le Yin

In Astronomy and Astrophysics

1. Huaihuo Cao
2. Binglong Chen
3. Xiping Zhu
4. S. Brendle
5. Le Yin
6. Peter Petersen
7. Xiaohua Zhu
8. B.-l. Chen

In Geometry and Topology

1. Huaihuo Cao
2. Binglong Chen
3. Xiping Zhu
4. S. Brendle
5. Peter Petersen
6. J. Lott
7. Le Yin
8. Siu-Hung Tang

Citations in the corpus

• Ricci flow with surgery on manifolds with positive isotropic curvature (2017, background): “For n = 4, this was proved in a landmark paper by Hamilton [20] (see also [11],[12]).”
• Manifolds with 1/4-pinched curvature are space forms (2007, background): “In order to extend the flow beyond singularities, Hamilton introduced the notion of Ricci flow with surgeries (see also [ 5 ], [24], [25]).”
• Recent developments on the Hamilton’s Ricci Flow (2007, methodology): “But this is a contradiction with the following basic geometry lemma, which was written down by the last two authors in [25].”
• Strong Uniqueness of the Ricci Flow (2007, methodology): “For an interesting application of this theorem to the theory of Ricci flow with surgery, we refer the readers to see [5] or relevant discussions in [1][10][11].”
• Uniqueness of the Ricci flow on complete noncompact manifolds (2005, methodology): “In [ 2 ], we observed that the condition of injectivity radius bounded from below ensures certain uniform (local) convexity and showed that this is sufficient to give the short time existence and the a priori estimates for the harmonic map flow.”
• Uniqueness and pseudolocality theorems of the mean curvature flow (2007, methodology): “We refer the readers to see an interesting application of this uniqueness theorem to the theory of the Ricci flow with surgery in dimension 3 and 4 [2, 4, 21].”
• Complete classification of compact four-manifolds with positive isotropic curvature (2008, methodology): “2 in [5] that the universal cover of X is S(3) ×R.”
• Isotropic Curvature and the Ricci Flow (2010, methodology): “This difficult result is proved using the Ricci flow with surgery; see also [4].”
• The Existence of Type II Singularities for the Ricci Flow on $S^{n+1}$ (2007, background): “In views of the work [23] of Hamilton on four-manifolds with positive isotropic curvature (see also [10]), a Type I singularity with neckpinch is expected.”
• Geometrisation of 3-Manifolds (2010, background): “[61, Section 2], [44, Section 61-64], and [17],[48] for the uniqueness.”
• New curvature conditions for the Bochner Technique (2019, methodology): “Based onRicci flowwith surgery, compactmanifoldswith positive isotropic curvature have been classified by Hamilton [22], Chen–Zhu [15] and Chen– Tang–Zhu [14] in dimension n = 4 and by Brendle [8] and Huang [24] in dimensions n ≥ 12.”