Sharp hyperbolic W1 comparison for the 2-capacity in dimensions n≥3

Establish the sharp comparison \(\mathrm{Cap}_2(K)\leq \mathrm{Cap}_2(\bar B(\mathsf r_1))\) for compact domains \(K\subset\mathbb H^n\) with smooth, star-shaped, and mean-convex boundary when \(n\geq 3\), where \(W_1(\bar B(\mathsf r_1))=W_1(K)\).

Background

The paper derives, for compact domains in hyperbolic space with smooth, star-shaped, and mean-convex boundary, a sharp comparison between the 2-capacity and a geodesic ball in dimension two. The comparison uses the area-matching radius $\mathsf r_1$, defined by $W_1(\bar B(\mathsf r_1))=W_1(K)$.

For dimensions $n\geq 3$, the authors explicitly state that the analogous sharp $W_1$-comparison is not currently available. Instead, the paper obtains the weaker estimate in equation (cap-Hn2a-high), which bounds the normalized capacity by a ratio involving $W_2(K)+n^{-1}|\bar B(\mathsf r_1)|$.