The Boundary between Compact and Noncompact Complete Riemann Manifolds
  In 1941 Sumner Myers proved that if the Ricci curvature of a complete Riemann manifold has a positive infimum then the manifold is compact and its diameter is bounded in terms of the infimum. Subsequently the curvature hypothesis has been weakened, and in this paper we weaken it further in an attempt to find the ultimate, sharp result. 
D. Holcman, C. Pugh