Smooth diameter and eigenvalue rigidity in positive Ricci curvature
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Abstract:
A recent injectivity radius estimate and previous sphere theorems yield the following smooth diameter sphere theorem for manifolds of positive Ricci curvature: For any given $m$ and $C$ there exists a positive constant $\varepsilon =\varepsilon (m,C)>0$ such that any $m$-dimensional complete Riemannian manifold with Ricci curvature $Ricc\ge m-1$, sectional curvature $K\le C$ and diameter $\ge \pi -\varepsilon$ is Lipschitz close and diffeomorphic to the standard unit $m$-sphere. A similar statement holds when the diameter is replaced by the first eigenvalue of the Laplacian.References
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Additional Information
- Wilderich Tuschmann
- Affiliation: Max-Planck-Institute for Mathematics in the Sciences, Inselstrasse, D-04103 Leipzig, Germany
- MR Author ID: 350718
- Email: tusch@mis.mpg.de
- Received by editor(s): October 26, 2000
- Published electronically: July 31, 2001
- Communicated by: Wolfgang Ziller
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 303-306
- MSC (2000): Primary 53C20
- DOI: https://doi.org/10.1090/S0002-9939-01-06384-5
- MathSciNet review: 1855649