# Volumes of balls in large Riemannian manifolds

```@article{Guth2006VolumesOB,
title={Volumes of balls in large Riemannian manifolds},
author={Larry Guth},
journal={Annals of Mathematics},
year={2006},
volume={173},
pages={51-76}
}```
• L. Guth
• Published 6 October 2006
• Mathematics
• Annals of Mathematics
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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