Relations of Four Asymptotic Geometric Quantities in Riemannian Geometry

TL;DR

This paper explores the asymptotic behavior of key geometric quantities on noncompact Riemannian manifolds, establishing inequalities and conditions under which these quantities coincide or differ.

Contribution

It introduces new asymptotic invariants and inequalities relating capacity, eigenvalues, and volume entropy, providing a unified perspective on manifold geometry.

Findings

The quantities ap, mbda, and ta are related by a general inequality involving volume entropy.

Under certain geometric conditions, these quantities coincide and equal the volume entropy or dimension.

Examples demonstrate that strict inequalities can occur, showing the bounds are sharp.

Abstract

This paper studies the large $p$ asymptotics of three geometric quantities on complete noncompact Riemannian manifolds: the $p-$capacity of a compact set, the first Dirichlet $p-$eigenvalue, and the Maz'ya constant, thereby offering a new perspective on the study of such manifolds. We introduce the infinity capacity $\mathcal{C}(\Omega)$, the infinity eigenvalue $\Lambda(M)$, and the Maz'ya limit $\mathcal{M}(M)$, and establish the general inequality, for any $\Omega\subset M$, $$ \mathcal{V}(M) \ge \mathcal{C}(\Omega) \ge \Lambda(M) = \mathcal{M}(M), $$ where $\mathcal{V}(M)$ is the volume entropy. Under geometric conditions such as isoperimetric control of balls, rotational symmetry, or curvature bounds, these quantities coincide and equal $\mathcal{V}(M)$ or the dimension. We also provide examples showing strict inequalities hold.