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Volumes of balls in large Riemannian manifolds. (English) Zbl 1232.53032

The author proves two important lower bounds for the volumes of balls in a Riemannian manifold:
(1) For each dimension \(n\), there is a number \(\delta(n)>0\) such that the following estimate holds: If \((M^n,g)\) is a complete Riemannian manifold with filling radius at least \(R\), then it contains a ball of radius \(R\) and volume at least \(\delta(n)R^n\).
(2) For each dimension \(n\), there is a number \(\delta(n)>0\) such that the following estimate holds: If \((M^n,\text{hyp})\) is a closed hyperbolic manifold and if \(g\) is another metric on \(M\) with volume not greater than \(\delta(n)\text{Vol}(M,\text{hyp})\), 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.

MSC:

53C20 Global Riemannian geometry, including pinching
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