论文标题
增强的主教 - 格罗莫夫定理
Enhanced Bishop-Gromov Theorem
论文作者
论文摘要
主教Gromov定理在空间中大地球体积的生长速率与RICCI曲率最负的成分有关。在本文中，我们证明了主教格罗莫夫（Gromov）的加强是统一空间。与原始的主教 - 格罗莫夫结合不同，我们增强的结合不仅取决于Ricci曲率的最负分量，还取决于整个光谱。进一步的结果，对于有限体积不均匀的空间，我们证明了在所有起点上的平均地球生长速率上的平均生长速率。该界限比从主教 - 格罗莫夫定理的界限强。我们的证明利用了Raychaudhuri方程，这是一个事实，即测量流程保守了相位空间的体积，也是我们引入的工具，用于研究我们称之为“系数改组”的相关雅各比方程的家族。
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".
