Curvature-Based Bounds for p-Capacity and Capacity--Area Inequalities
Abstract
We establish curvature-based bounds for the p-capacity of convex domains and related capacity--area inequalities. Let K denote the Gauss--Kronecker curvature of ∂Ω. In Euclidean space, if $ 0<K_-\leq K\leq K_+, $ then, for every 1<p≤ (n+1)/2, Capp(B(K+1/(1−n))) ≤ Capp(Ω) ≤ Capp(B(K-1/(1−n))). The lower comparison remains valid for the full range 1<p<n. Moreover, the exponent range in the upper comparison is optimal in general: when n≥ 4 and (n+1)/2<p<n, the upper comparison may fail. Thus p=(n+1)/2 is the critical exponent for the curvature-radius upper comparison. Under the pinching condition $ \frac{K_-}{K_+}=ρ\in (0, 1], $ we further obtain an explicit scale-invariant capacity--area lower bound for every 1<p<n. In the special case n=3 and p=2, Cap2(Ω)/√(|∂Ω|) ≥ (4√π ρ1/4)/(1+√ρ), which implies the Pólya--Szegö inequality whenever ρ≥ ρ*:= ( (π2−4−π√(π2−8))/4 )2 ≃ 0.15484. On a Cartan--Hadamard manifold with Sec≤ −k2<0, we obtain a p-capacity lower bound retaining all principal curvatures of the boundary, together with corresponding Gauss--Kronecker curvature estimates and a rigidity characterization of the equality case by a warped-product exterior geometry.
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BibTeX
@misc{jin2026curvaturebased,
title = {{Curvature-Based Bounds for $p$-Capacity and Capacity--Area Inequalities}},
author = {Xiaoshang Jin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07564},
eprint = {2610.07564},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07564},
}
RIS
TY - UNPB
TI - Curvature-Based Bounds for p-Capacity and Capacity--Area Inequalities
AU - Xiaoshang Jin
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.07564
UR - https://arxiv.org/abs/2610.07564
AB - We establish curvature-based bounds for the p-capacity of convex domains and related capacity--area inequalities. Let K denote the Gauss--Kronecker curvature of ∂Ω. In Euclidean space, if $ 0<K_-\leq K\leq K_+, $ then, for every 1<p≤ (n+1)/2, Cap_p(B(K_+^1/(1−n))) ≤ Cap_p(Ω) ≤ Cap_p(B(K_-^1/(1−n))). The lower comparison remains valid for the full range 1<p<n. Moreover, the exponent range in the upper comparison is optimal in general: when n≥ 4 and (n+1)/2<p<n, the upper comparison may fail. Thus p=(n+1)/2 is the critical exponent for the curvature-radius upper comparison. Under the pinching condition $ \frac{K_-}{K_+}=ρ\in (0, 1], $ we further obtain an explicit scale-invariant capacity--area lower bound for every 1<p<n. In the special case n=3 and p=2, Cap_2(Ω)/√(|∂Ω|) ≥ (4√π ρ^1/4)/(1+√ρ), which implies the Pólya--Szegö inequality whenever ρ≥ ρ_*:= ( (π^2−4−π√(π^2−8))/4 )^2 ≃ 0.15484. On a Cartan--Hadamard manifold with Sec≤ −k^2<0, we obtain a p-capacity lower bound retaining all principal curvatures of the boundary, together with corresponding Gauss--Kronecker curvature estimates and a rigidity characterization of the equality case by a warped-product exterior geometry.
ER -