Paper · Aug 18, 2026

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

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Abstract

This paper establishes sharp upper bounds for $p$ -capacities $Cap_{1 < p < \infty}$ in the hyperbolic space $H^{n}$ 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 $W_{1} ∣ W_{2}$ . 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 $p > 2 m + 1$ , an interpolating radius combines the $2 m$ -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.

Comments: 30 pages

Mean curvature flow Other curvature flows Inverse mean curvature flow Geometric inequalities

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Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Inverse mean curvature flow Wider flows A hypersurface expands with normal speed 1/H, the reciprocal of its mean curvature. Huisken and Ilmanen's weak version keeps the Hawking mass… Start with: Flow of nonconvex hypersurfaces into spheres (Gerhardt, 1990) Geometric inequalities Sharp inequalities between area, volume and integrals of curvature, with equality only for round spheres. Start with: Mean curvature flow and geometric inequalities (Topping, 1998)

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6A.10
Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives
Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.

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

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Sharp p-Capacity Estimates via Quermassintegrals in Hyperbolic Space

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