Volumes of Balls in Large Riemannian Manifolds

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 (Mn, hyp) is a closed hyperbolic manifold and if g is another metric on M with volume at most δ(n)V ol(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. Let (M, g) be a Riemannian manifold of dimension n. Let V (R) denote the largest volume of any metric ball of radius R in (M, g). In [4], Gromov made a number of conjectures relating the function V (R) to other geometric invariants of (M, g). The spirit of these conjectures is that if (M, g) is “large”, then V (R) should also be large. In this paper, we prove some of Gromov’s conjectures. Our first result involves the filling radius of (M, g), defined in [5]. Roughly speaking, the filling radius describes how “thick” a Riemannian manifold is. For example, the standard product metric on the cylinder S × R has filling radius π/2, and the Euclidean metric on R has infinite filling radius. Theorem 1. For each dimension n, there is a number δ(n) > 0 so that the following estimate holds. If (M, g) is a complete Riemannian n-manifold with filling radius at least R, then V (R) ≥ δ(n)R. Our second result involves a closed hyperbolic manifold (M,hyp) equipped with an auxiliary metric g. Slightly paradoxically, if the manifold (M, g) is small, then its universal cover (M̃, g̃) tends to be large. For example, if we look at the universal cover of (M,λhyp), we get the space form with constant curvature −λ. As λ decreases, the strength of the curvature increases, which increases the volumes of balls. Our second theorem gives a large ball in the universal cover (M̃, g̃) provided that the volume of (M, g) is sufficiently small. Theorem 2. For each dimension n, there is a number δ(n) > 0 so that the following estimate holds. Suppose that (M, hyp) is a closed hyperbolic n-manifold and that g is another metric on M , and suppose that V ol(M, g) < δ(n)V ol(M,hyp). Let (M̃, g̃) denote the universal cover of M with the metric induced from g. Then there is a point p ∈ M̃ so that the unit ball around p in (M̃, g̃) has a larger volume than the unit ball in hyperbolic n-space. In other words, the following inequality holds. V(M̃,g̃)(1) > VHn(1). We spend most of this introduction giving a context for these two results. At the end, we give a quick overview of the proof. Many readers may not be familiar with the filling radius. Before looking at its definition, we give some corollaries of Theorem 1 using more common vocabulary.