Integral curvature bounds and bounded diameter. (English) Zbl 0984.53018

Sprouse, Chadwick

Commun. Anal. Geom. 8, No. 3, 531-543 (2000).

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.

Reviewer: Lew Friedland (Geneseo)

MSC:

53C21

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

Keywords:

Riemannian manifolds; integral curvature bounds; Ricci curvature; fundamental group; lowest eigenvalue of the Ricci tensor