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

Shouhei Honda

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

Differential Geometry 2014-03-04 v3 Analysis of PDEs Metric Geometry

Authors: Shouhei Honda

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.

Keywords

Cheeger constant p-Laplacian Gromov-Hausdorff convergence Riemannian manifold first eigenvalue Ricci curvature isoperimetric inequalities Lichnerowicz-Obata theorem

Cite

@article{arxiv.1310.0304,
  title  = {Cheeger constant, $p$-Laplacian, and Gromov-Hausdorff convergence},
  author = {Shouhei Honda},
  journal= {arXiv preprint arXiv:1310.0304},
  year   = {2014}
}

Comments

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

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

Abstract

Keywords

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