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Volumes of balls in large Riemannian manifolds

Larry Guth

TL;DR

The paper proves two Gromov-inspired results: (1) any complete n-manifold with filling radius at least R must contain a ball of radius R and volume at least c(n)R^n, and (2) for closed hyperbolic M with Vol(M,g) small relative to Vol(M,hyp), the universal cover contains a unit ball larger than in hyperbolic space. The authors develop a framework of good balls, a rectangular nerve, and high-multiplicity-volume controls to transfer geometric thickness into lower bounds on ball volumes, using filling-radius arguments and simplicial norms. The approach yields corollaries such as systolic-type inequalities and hyperbolic-volume comparisons, and extends to open manifolds with adjustments. They also pose open questions on relating volume to Uryson width, suggesting further avenues for geometric-analytic bounds via nerve-based methods.

Abstract

If (M^n, g) is a complete Riemannian manifold with filling radius at least R, then we prove that it contains a ball of radius R and volume at least c(n)R^n. If (M^n, hyp) is a closed hyperbolic manifold and if g is another metric on M with volume at most c(n)Volume(M,hyp), then we prove that the universal cover of (M,g) contains a unit ball with volume greater than the volume of a unit ball in hyperbolic n-space.

Volumes of balls in large Riemannian manifolds

TL;DR

The paper proves two Gromov-inspired results: (1) any complete n-manifold with filling radius at least R must contain a ball of radius R and volume at least c(n)R^n, and (2) for closed hyperbolic M with Vol(M,g) small relative to Vol(M,hyp), the universal cover contains a unit ball larger than in hyperbolic space. The authors develop a framework of good balls, a rectangular nerve, and high-multiplicity-volume controls to transfer geometric thickness into lower bounds on ball volumes, using filling-radius arguments and simplicial norms. The approach yields corollaries such as systolic-type inequalities and hyperbolic-volume comparisons, and extends to open manifolds with adjustments. They also pose open questions on relating volume to Uryson width, suggesting further avenues for geometric-analytic bounds via nerve-based methods.

Abstract

If (M^n, g) is a complete Riemannian manifold with filling radius at least R, then we prove that it contains a ball of radius R and volume at least c(n)R^n. If (M^n, hyp) is a closed hyperbolic manifold and if g is another metric on M with volume at most c(n)Volume(M,hyp), then we prove that the universal cover of (M,g) contains a unit ball with volume greater than the volume of a unit ball in hyperbolic n-space.

Paper Structure

This paper contains 7 sections, 24 theorems, 36 equations.

Table of Contents

  1. Good balls and good covers
  2. The volume of the high-multiplicity set
  3. The rectangular nerve
  4. Filling cycles in the rectangular nerve
  5. Estimates for simplicial norms
  6. Open manifolds
  7. A question about Uryson width

Key Result

Theorem 1

For each dimension n, there is a number $\delta(n) > 0$ so that the following estimate holds. If $(M^n,g)$ is a complete Riemannian n-manifold with filling radius at least $R$, then $V(R) \ge \delta(n) R^n$.

Theorems & Definitions (28)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Conjecture
  • ...and 18 more