AI 中文总结
研究具有Ricci曲率下界的完整非紧黎曼流形上p-容量的精确估计，通过建立比较不等式及刻画等号成立情况得到结果，还研究了电容器相对p-容量的归一化下界，引入尺度不变量确定其最优范围。
AI 中文摘要
我们研究在Ricci曲率下界下完整非紧黎曼流形上p-容量的精确估计。首先，我们为满足Ric≥ -ng的流形中有界光滑区域的p-容量建立精确比较不等式。这些估计用边界平均曲率表示，并对应于自然的扭曲积模型端。我们刻画了所有等号成立的情况，并表明等号成立迫使外部区域与相应的扭曲积等距。我们还在非负Ricci曲率下得到了类似的精确估计，其等号成立情况由一个渐近平坦模型端描述。其次，我们研究电容器相对p-容量的归一化下界。我们引入涉及内部集合体积和环境域直径的尺度不变量，建立一致的正下界，并确定归一化参数的最优范围。
英文摘要
We study sharp estimates for the $p$-capacity on complete non-compact Riemannian manifolds under lower Ricci curvature bounds. First, we establish sharp comparison inequalities for the $p$-capacity of bounded smooth domains in manifolds satisfying $\operatorname{Ric}\ge -ng.$ The estimates are expressed in terms of the boundary mean curvature and correspond to natural warped-product model ends. We characterize all equality cases and show that equality forces the exterior region to be isometric to the corresponding warped product. We also obtain an analogous sharp estimate under nonnegative Ricci curvature, whose equality case is described by an asymptotically flat model end. Second, we investigate normalized lower bounds for the relative $p$-capacity of condensers. We introduce scale-invariant quantities involving the volume of the inner set and the diameter of the ambient domain, establish uniform positive lower bounds, and determine the optimal ranges of the normalization parameters.
Comments22 pages, comments are welcome
