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Curvature, Sphere Theorems, and the Ricci flow


Authors: Simon Brendle and Richard Schoen
Journal: Bull. Amer. Math. Soc. 48 (2011), 1-32
MSC (2010): Primary 53C21; Secondary 53C20, 53C24, 53C43, 53C44, 35K55
DOI: https://doi.org/10.1090/S0273-0979-2010-01312-4
Published electronically: September 29, 2010
MathSciNet review: 2738904
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Abstract: In 1926, Hopf proved that any compact, simply connected Riemannian manifold with constant curvature $ 1$ is isometric to the standard sphere. Motivated by this result, Hopf posed the question whether a compact, simply connected manifold with suitably pinched curvature is topologically a sphere.

In the first part of this paper, we provide a background discussion, aimed at nonexperts, of Hopf's pinching problem and the Sphere Theorem. In the second part, we sketch the proof of the Differentiable Sphere Theorem, and discuss various related results. These results employ a variety of methods, including geodesic and minimal surface techniques as well as Hamilton's Ricci flow.


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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/S0273-0979-2010-01312-4
Received by editor(s): January 13, 2010
Published electronically: September 29, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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