Cheeger constant, $p$-Laplacian, and Gromov-Hausdorff convergence

Abstract

We discuss the behavior of $({\lambda }_{1.p}(M){)}^{1/p}$ with respect to the Gromov-Hausdorff topology and the variable $p$, where ${\lambda }_{1,p}(M)$ is the first positive eigenvalue of the $p$-Laplacian on a compact Riemannian manifold $M$. Applications include new estimates for the first eigenvalues of the $p$-Laplacian on Riemannian manifolds with lower Ricci curvature bounds, and isoperimetric inequalities on Gromov-Hausdorff limit spaces. We also establish a new Lichnerowicz-Obata type theorem.

25 pages. A main result of the previous version is changed

References in corpus (4)

• Sharp estimates on the first eigenvalue of the p-Laplacian with negative Ricci lower bound
• Local Poincaré inequalities from stable curvature conditions on metric spaces
• A weakly second order differential structure on rectifiable metric measure spaces
• Eigenvalues of Laplacian and multi-way isoperimetric constants on weighted Riemannian manifolds

Cited by in corpus (5)

• Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds
• Spectral convergence under bounded Ricci curvature
• Convergence of continuous stochastic processes on compact metric spaces converging in the Lipschitz distance
• Certain min-max values related to the $p$-energy and packing radii of Riemannian manifolds and metric measure spaces
• Principal eigenvalue problem for infinity Laplacian in metric spaces