## 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.

### MSC:

 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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