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

2019/12/03 by Ayato Mitsuishi, Mitsuishi, Ayato
Mathematics · Physics and Astronomy · #53C20 #53C23 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1912.01432

openalex publication_date 2019/12/03 · openalex created_date 2019/12/13 · openalex updated_date 2026/07/28

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 → ∞. 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.

Citations

• Cheeger constant, p-Laplacian, and Gromov-Hausdorff convergence
• Ricci curvature for metric-measure spaces via optimal transport
• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Local Poincaré inequalities from stable curvature conditions on metric spaces
• Lower Bounds on Ricci Curvature and the Almost Rigidity of Warped Products
• Quasiconformal maps in metric spaces with controlled geometry
• The splitting theorem in non-smooth context
• Infinitesimal Hilbertianity of locally CAT(κ)-spaces
• Local Poincar 'e inequalities from stable curvature conditions on metric\n spaces

Related

• The lower bound of first Dirichlet eigenvalue of p-Laplacian in Riemannian manifolds
• Asymptotics of the first Laplace eigenvalue with Dirichlet regions of prescribed length
• Conformal upper bounds for the first eigenvalue of the p-Laplacian
• Metric upper bounds for Steklov and Laplace eigenvalues
• Min-max harmonic maps and a new characterization of conformal eigenvalues