On complete noncompact Riemannian manifolds the volume entropy bounds the infinity capacity which bounds the infinity eigenvalue, which equals the Maz'ya limit, with equality under isoperimetric or curvature conditions.
Sharply estimating hyperbolic capacities
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Relations of Four Asymptotic Geometric Quantities in Riemannian Geometry
On complete noncompact Riemannian manifolds the volume entropy bounds the infinity capacity which bounds the infinity eigenvalue, which equals the Maz'ya limit, with equality under isoperimetric or curvature conditions.