Volumes of balls in large riemannian manifolds

Abstract

We prove two lower bounds for the volumes of balls in a Riemannian manifold. If (Mn,g) is a complete Riemannian manifold with filling radius at least R, then it contains a ball of radius R and volume at least δ(n)Rn. If (M n, hyp) is a closed hyperbolic manifold and if g is another metric on M with volume no greater than δ(n)Vol(M, hyp), then the universal cover of (M, g) contains a unit ball with volume greater than the volume of a unit ball in hyperbolic n-space.