论文标题
平均标态曲率对Riemannian流形的影响
Effect of the average scalar curvature on Riemannian manifolds
论文作者
论文摘要
我们研究了平均标态曲率对结合半径的影响，测量球的平均面积，平均公制球的平均体积以及封闭的Riemannian歧管$ n $的总体积（或更通常的$ n $带有有限量，其负ricci curvator curvation curvature curvature curvature curvature in $ sn $是有限的）。例如，我们证明，如果平均标量曲率大于归一化RICCI曲率的下限，那么我们可以根据任何大小的公制球的平均体积来改善主教 - 格罗莫夫估计值。当平均标量曲率具有下限时，我们还证明了某个几何积分的单调降低特性。这导致了比较的定理，即半径高达$ \ mathrm {inj}（n）$的平均总平均曲率。
We investigate the effect of the average scalar curvature on the conjugate radius, average area of the geodesic spheres, average volume of the metric balls and the total volume of a closed Riemannian manifold $N$ (or more generally $N$ with finite volume whose negative Ricci curvature integral on $SN$ is finite). For example, we prove that if the average scalar curvature is larger than the lower bound of the normalized Ricci curvature, then we can improve the Bishop-Gromov estimate on the average volume of the metric balls of any size. We also prove the monotone decreasing property of a certain geometric integral when the average scalar curvature has a lower bound. This leads to a comparison theorem of the average total mean curvature of geodesic spheres of radius up to $\mathrm{inj}(N)$.
