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Integral curvature bounds and bounded diameter. (English) Zbl 0984.53018
The paper generalizes the result of Myers that a complete $n$-dimensional Riemannian manifold $( M,g) $ with Ricci curvature $\text{Ric}(M) \geq ( n-1) k>0$ is compact, has $\text{diam}( M)\leq \pi /\sqrt{k}$ and fundamental group $\pi _1( M) $ finite by using an inequality of Cheeger and Colding. The author proves that if $( M,g) $ is compact and $\text{Ric}( M) \geq 0,$ given $\delta >0$ and a sufficiently small (depending on $\delta $ and $n)$ integral involving the lowest eigenvalue of the Ricci tensor, then $\text{diam}( M) <\pi +\delta $. If $M$ is either not compact or the Ricci curvature is bounded nonpositively below, that is, $\text{Ric}( M)\geq ( n-1) k$ with $k\leq 0$ but satisfies a similar integral curvature bound, then $M$ is compact with bounded diameter and finite fundamental group. The integral curvature condition replaces bounds on $\text{diam}( M) $ from above and $\text{vol}( M) $ from below as were assumed in previous extensions of Myers’ result that $\pi _1(M) $ is finite.

53C21Methods of Riemannian geometry, including PDE methods; curvature restrictions (global)