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Smooth diameter and eigenvalue rigidity in positive Ricci curvature

Author(s): Wilderich Tuschmann
Journal: Proc. Amer. Math. Soc. 130 (2002), 303-306.
MSC (2000): Primary 53C20
Posted: July 31, 2001
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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.

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Additional Information:

Wilderich Tuschmann
Affiliation: Max-Planck-Institute for Mathematics in the Sciences, Inselstrasse, D-04103 Leipzig, Germany
Email: tusch@mis.mpg.de

DOI: 10.1090/S0002-9939-01-06384-5
PII: S 0002-9939(01)06384-5
Keywords: Sphere theorems, injectivity radius, exotic spheres, positive Ricci curvature
Received by editor(s): October 26, 2000
Posted: July 31, 2001
Communicated by: Wolfgang Ziller
Copyright of article: Copyright 2001, American Mathematical Society

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