Curvature, Sphere Theorems, and the Ricci flow
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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
- MR Author ID: 655348
- Richard Schoen
- Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
- Received by editor(s): January 13, 2010
- Published electronically: September 29, 2010
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2738904