A general convergence result for the Ricci flow in higher dimensions

By S. Brendle. 2007

113 citations recorded by the source corpus.

Let (M,g_0) be a compact Riemannian manifold of dimension n \geq 4. We show that the normalized Ricci flow deforms g_0 to a constant curvature metric provided that (M,g_0) x R has positive isotropic curvature. This condition is stronger than 2-positive flag curvature but weaker than 2-positive curvature operator.

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

View Paper: https://www.semanticscholar.org/paper/78c7b23c9840c72bd22472c2dc92afe1f2d3de9e

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. Hong-wei Xu
2. J. Gu
3. Lu Wang
4. Brett Kotschwar
5. Burkhard Wilking
6. S. Brendle
7. Entao Zhao
8. Robert Haslhofer

In Astronomy and Astrophysics

1. Burkhard Wilking
2. Hong-wei Xu
3. J. Gu
4. Lu Wang
5. Brett Kotschwar
6. Entao Zhao
7. S. Brendle
8. Peter Petersen

In Geometry and Topology

1. Hong-wei Xu
2. J. Gu
3. Lu Wang
4. Burkhard Wilking
5. Brett Kotschwar
6. Entao Zhao
7. S. Brendle
8. Peter Petersen

Citations in the corpus

• Topological and differentiable sphere theorems for complete submanifolds (2009, background): “More recently, Brendle [12] obtained the following useful result.”
• The sphere theorems for manifolds with positive scalar curvature (2011, result): “[8] S. Brendle and R. Schoen, Sphere theorems in geometry, Surveys in Differential Geometry, Vol.13, 2009, 49-84.”
• Ricci flow with surgery on manifolds with positive isotropic curvature (2017, background): “For an initial metric that is strictly PIC1, it was shown in [2] that the Ricci flow will converge to a metric of constant curvature after rescaling (see [1],[8],[21],[25],[26],[27],[30] for earlier work on the subject).”
• A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities (2010, methodology): “2 follows from Brendle [2009]. He derived a similar tensor identity for Hm using the following torsion free connection that is similar to the one introduced by Chow and Chu [1995].”
• Geometric, topological and differentiable rigidity of submanifolds in space forms (2011, methodology): “After the pioneering rigidity theorem for minimal submanifolds in a sphere due to Simons [21], Lawson [15] and Chern-do Carmo-Kobayashi [9], Yau [31] and Ejiri [10] obtained three important rigidity theorems for oriented compact minimal submanifolds in Sn+p.”
• Rigidity of asymptotically conical shrinking gradient Ricci solitons (2013, result): “In a later paper, following the same general outline, Brendle [7] extended his theorem to higher-dimensional steady solitons of positive curvature operator which blow-down similarly to a cylinder.”