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Journal of the American Mathematical Society

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Manifolds with $ 1/4$-pinched curvature are space forms


Authors: Simon Brendle and Richard Schoen
Journal: J. Amer. Math. Soc. 22 (2009), 287-307
MSC (2000): Primary 53C20; Secondary 53C44
DOI: https://doi.org/10.1090/S0894-0347-08-00613-9
Published electronically: July 17, 2008
MathSciNet review: 2449060
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Abstract: 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 Böhm and Wilking.


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Additional Information

Simon Brendle
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305

Richard Schoen
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305

DOI: https://doi.org/10.1090/S0894-0347-08-00613-9
Keywords: Ricci flow, curvature pinching, sphere theorem
Received by editor(s): May 7, 2007
Published electronically: July 17, 2008
Additional Notes: The first author was partially supported by a Sloan Foundation Fellowship and by NSF grant DMS-0605223
The second author was partially supported by NSF grant DMS-0604960.
Article copyright: © Copyright 2008 American Mathematical Society