Comparison results for capacity

TL;DR

This paper establishes bounds for the capacity of compact sets in various Riemannian manifolds based on curvature conditions, providing characterizations of equality cases and extending results to convex sets in Euclidean space.

Contribution

It derives new capacity bounds under curvature constraints and characterizes the cases of equality, extending classical geometric inequalities to broader settings.

Findings

Capacity bounds depend on boundary curvature and ambient manifold curvature.

Equality cases are characterized by geometric symmetry, such as spheres.

Results extend classical Euclidean inequalities to curved spaces.

Abstract

We obtain in this paper bounds for the capacity of a compact set $K$. If $K$ is contained in an $(n+1)$-dimensional Cartan-Hadamard manifold, has smooth boundary, and the principal curvatures of $\partial K$ are larger than or equal to $H_0>0$, then ${\rm Cap}(K)\geq (n-1)\,H_0{\rm vol}(\partial K)$. When $K$ is contained in an $(n+1)$-dimensional manifold with non-negative Ricci curvature, has smooth boundary, and the mean curvature of $\partial K$ is smaller than or equal to $H_0$, we prove the inequality ${\rm Cap}(K)\leq (n-1)\,H_0{\rm vol}(\partial K)$. In both cases we are able to characterize the equality case. Finally, if $K$ is a convex set in Euclidean space $\mathbb{R}^{n+1}$ which admits a supporting sphere of radius $H_0^{-1}$ at any boundary point, then we prove ${\rm Cap}(K)\geq (n-1)\,H_0\mathcal{H}^n(\partial K)$ and that equality holds for the round sphere of radius $H_0^{-1}$.