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In 1941 Sumner Myers proved that if the Ricci curvature of a complete Riemannmanifold has a positive infimum then the manifold is compact and its diameteris bounded in terms of the infimum. Subsequently the curvature hypothesis hasbeen weakened, and in this paper we weaken it further in an attempt to find theultimate, sharp result.

The boundary between compact and noncompact complete Riemann manifolds