Volumes of small balls on open manifolds: lower bounds and examples

Abstract:

Question: "Under what curvature assumptions on a complete open manifold is the volume of balls of a fixed radius bounded below independent of the center point?" Two theorems establish such assumptions and two examples sharply limit their weakening. In particular we give an example of a metric on ${{\mathbf {R}}^4}$ (extending to higher dimensions) of positive Ricci curvature, whose sectional curvatures decay to $0$, and such that the volume of balls goes uniformly to $0$ as the center goes to infinity.