Certain min-max values related to the $p$-energy and packing radii of Riemannian manifolds and metric measure spaces

TL;DR

This paper extends known results about the asymptotic behavior of eigenvalues related to the $p$-energy on Riemannian manifolds, connecting them to packing radii and applicable in more singular settings.

Contribution

It generalizes the convergence results of eigenvalues to packing radii for higher min-max values and broadens applicability to more singular metric measure spaces.

Findings

Convergence of $k$-th min-max values to packing radii as $p o rac{1}{p}$

Extension of results to more singular metric measure spaces

Connection between eigenvalues and geometric packing measures

Abstract

Grosjean proved that the $(1/p)$-th power of the first eigenvalue of the $p$-Laplacian on a closed Riemannian manifold converges to the twice of the inverse of the diameter of the space, as $p \to \infty$. Before this, a corresponding result for the Dirichlet first eigenvalues was also obtained by Juutinen, Lindqvist and Manfredi. We extend those results for certain $k$-th min-max value related to the $p$-energy, where the corresponding limits are packing radii introduced by Grove-Markvorsen or its variant. Furthermore, we remark that our result holds for more singular setting.