Sharp $p$-Capacity Estimates via Quermassintegrals in Hyperbolic Space

Abstract

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.

30 pages