Manifolds with 1/4-pinched curvature are space forms

By S. Brendle, R. Schoen. 2007

394 citations recorded by the source corpus.

Let (M,g_0) be a compact Riemannian manifold with pointwise 1/4-pinched sectional curvatures. We show that the Ricci flow deforms g_0 to a constant curvature metric. The proof uses the fact, also established in this paper, that positive isotropic curvature is preserved by the Ricci flow in all dimensions. We also rely on earlier work of Hamilton and of Bohm and Wilking.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6fe3e0584189ffff552677d8f30b43bd4d19d5f8

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In Applied Mathematics

1. Hong-wei Xu
2. Burkhard Wilking
3. J. Gu
4. Song Sun
5. S. Brendle
6. Lu Wang
7. Brett Kotschwar
8. Weiyong He

In Geometry and Topology

1. Hong-wei Xu
2. Burkhard Wilking
3. J. Gu
4. Song Sun
5. Lu Wang
6. S. Brendle
7. Weiyong He
8. Esther Cabezas-Rivas

Citations in the corpus

• A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities (2010, result): “This recovers the main theorem of [Brendle, 2008], which in turn generalized the main result of [Brendle and Schoen, 2009].”
• Rigidity of asymptotically conical shrinking gradient Ricci solitons (2013, result): “The results of Böhm-Wilking [2], for example, imply that two-positivity of the curvature operator is sufficient, and the results of Brendle-Schoen [8] and Brendle [5] demonstrate that this can be relaxed further still, e.”
• Topological and differentiable sphere theorems for complete submanifolds (2009, methodology): “Recently, Brendle and Schoen [10, 11] proved a remarkable differentiable pinching theorem for point-wise 1/4-pinched Riemannian manifolds by developing the theory and techniques of Ricci flow [22–24].”
• Frankel conjecture and Sasaki geometry (2012, methodology): “There are lots of recent deep progress on this, especially using the technique of Ricci flow, see [4] and [9] for example.”
• Classification of manifolds with weakly 1/4-pinched curvatures (2007, methodology): “If (M,g0)×R(2) has nonnegative isotropic curvature, then, by the results in [5], the product (M,g(t)) × R2 has nonnegative isotropic curvature for all t ∈ [0, T ).”
• The local entropy along Ricci flow---Part A: the no-local-collapsing theorems (2017, background): “Such developments finally lead to the sphere theorem of Brendle-Schoen [2], which asserts that starting from a manifold whose Riemannian curvature is quater-pinched, the normalized Ricci flow (1.”
• The sphere theorems for manifolds with positive scalar curvature (2011, methodology): “The Brendle-Schoen Differentiable Sphere Theorem [9, 10] brought us a big break through in the investigation of curvature and topology of manifolds.”
• Isotropic Curvature and the Ricci Flow (2010, methodology): “We would like to mention that the above theorem was independently proved in [3], where it was used to resolve the 1/4 pinching sectional curvature diffeomorphism sphere theorem.”
• Immortal homogeneous Ricci flows (2017, result): “ curvature can be deformed to the round metric via volume normalized Ricci flow [Ham82]. Similar convergence results were obtained afterwards in higher dimensions, see for instance [Ham86], [BW08] and [BS09]. About 20 years later, Perelman was able to describe new geometric quantities, which are mono”
• How to produce a Ricci Flow via Cheeger-Gromoll exhaustion (2011, background): “Here we collect some relevant features known about the above curvature condition (see [6] and [35] for the proofs).”