Abstract
We prove two lower bounds for the volumes of balls in a Riemannian manifold. If (M n ;g) is a complete Riemannian manifold with lling radius at least R, then it contains a ball of radius R and volume at least (n)R n . If (M n ; hyp) is a closed hyperbolic manifold and if g is another metric on M with volume no greater than (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.
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