Relations of Four Asymptotic Geometric Quantities in Riemannian Geometry

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.