An Improved Long-Time Bishop-Gromov Theorem Using Shear

Abstract

The Bishop-Gromov theorem is a comparison theorem of differential geometry that upperbounds the growth of volume of a geodesic ball in a curved space. For many spaces, this bound is far from tight. We identify a major reason the bound fails to be tight: it neglects the effect of shear. By using higher curvature invariants to lowerbound the average shear, we are able to place tighter-than-Bishop-Gromov upperbounds on the late-time growth rates of geodesic balls in homogeneous spaces with non-positive sectional curvature. We also provide concrete guidance on how our theorem can be generalized to inhomogeneous spaces, to spaces with positive sectional curvatures, and to intermediate and short times. In arXiv:2209.09288 we discovered an enhancement to the BG theorem that was strongest at early times, and that relied upon additive properties of families of Jacobi equations; in this paper we find a different enhancement at late times that connects to multiplicative properties of families of Jacobi equations. A novel feature shared by both papers is the consideration of families of equations that are not coupled but whose coefficients are correlated.