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Enhanced Bishop-Gromov Theorem

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”.