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. Finally, combining with the entropy rigidity theorem, we obtain a characterization of hyperbolic manifolds. We also provide examples showing strict inequalities hold.
From: Xiaoshang Jin [view email]
[v1]
Thu, 16 Apr 2026 04:11:19 UTC (15 KB)
[v2]
Fri, 29 May 2026 13:37:14 UTC (18 KB)
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