Smooth diameter and eigenvalue rigidity in positive Ricci curvature

Tuschmann, Wilderich

doi:10.1090/S0002-9939-01-06384-5

Smooth diameter and eigenvalue rigidity in positive Ricci curvature
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Proc. Amer. Math. Soc. 130 (2002), 303-306 Request permission

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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