AI 中文总结
该研究在双曲空间中通过Quermassintegral和有效曲率半径建立p-容量的精确上界，利用逆平均曲率流得到曲率半径，将估计转化为容量与面积比的测地球比较，等式对应测地球。
AI 中文摘要
本文通过双曲Quermassintegral和有效曲率半径，建立了双曲空间$\boldsymbol{\reals}^n$中$1<p<\boldsymbol{\reals}$的p-容量$\text{Cap}$的精确上界。Quermassintegral比较涉及$W_{n-1}$、$W_{k+1}+\frac{k(n+1-k)}{}W_{k-1}$以及$W_1|W_2$。对于星形、平均凸或h-超曲面，逆平均曲率流进一步产生由归一化平均曲率的$L^q$平均及其平方双曲过剩矩确定的曲率半径。这些半径将所得估计转换为容量与面积比的精确测地球比较。在$p>2m+1$范围内，插值半径结合了第$2m$阶曲率过剩半径与$L^\boldsymbol{\reals}$曲率尺度，从而连接有限矩和上确界 regime。精确比较中的等式表征了测地球。
英文摘要
This paper establishes sharp upper bounds for $p$-capacities $\mathrm{Cap}_{1<p<\infty}$ in the hyperbolic space $\mathbb{H}^n$ through hyperbolic quermassintegrals and effective curvature radii. The quermassintegral comparisons involve $W_{n-1}$, $W_{k+1}+k(n+1-k)^{-1}W_{k-1}$, and the pair $W_1\mid W_2$. For star-shaped, mean-convex or h-convex hypersurfaces, inverse mean curvature flow further produces curvature radii determined by $L^q$-averages of the normalized mean curvature and by moments of its squared hyperbolic excess. These radii convert the resulting estimates into sharp geodesic-ball comparisons for the capacity-to-area ratio. In the range $p>2m+1$, an interpolating radius combines the $2m$-th curvature-excess radius with the $L^\infty$ curvature scale, thereby linking the finite-moment and supremum regimes. Equality in the sharp comparisons characterizes geodesic balls.
Comments30 pages
