Enhanced Bishop-Gromov Theorem

TL;DR

This paper strengthens the Bishop-Gromov theorem for homogeneous spaces by incorporating the full Ricci curvature spectrum and provides new bounds on geodesic growth in inhomogeneous spaces, using advanced geometric analysis tools.

Contribution

It introduces a refined volume growth bound depending on the full Ricci spectrum and a new method called 'coefficient shuffling' for analyzing Jacobi equations.

Findings

Enhanced volume growth bounds for homogeneous spaces.

Stronger average geodesic growth bounds in inhomogeneous spaces.

Introduction of the 'coefficient shuffling' technique for Jacobi equations.

Abstract

The Bishop-Gromov theorem upperbounds the rate of growth of volume of geodesic balls in a space, in terms of the most negative component of the Ricci curvature. In this paper we prove a strengthening of the Bishop-Gromov bound for homogeneous spaces. Unlike the original Bishop-Gromov bound, our enhanced bound depends not only on the most negative component of the Ricci curvature, but on the full spectrum. As a further result, for finite-volume inhomogeneous spaces, we prove an upperbound on the average rate of growth of geodesics, averaged over all starting points; this bound is stronger than the one that follows from the Bishop-Gromov theorem. Our proof makes use of the Raychaudhuri equation, of the fact that geodesic flow conserves phase-space volume, and also of a tool we introduce for studying families of correlated Jacobi equations that we call "coefficient shuffling".