Ricci curvature and $L^p$-convergence

TL;DR

This paper introduces $L^p$-convergence of tensor fields in Gromov-Hausdorff topology, establishing fundamental properties, and applies it to derive inequalities, formulas, and eigenvalue continuity on limit spaces of Riemannian manifolds with Ricci bounds.

Contribution

It defines $L^p$-convergence in this setting and applies it to derive a Bochner-type inequality, a formula for the Dirichlet Laplacian, and eigenvalue continuity results.

Findings

Established $L^p$-convergence properties of tensor fields.

Derived a Bochner-type inequality on limit spaces.

Proved continuity of first eigenvalues of the p-Laplacian.

Abstract

We give the definition of $L^p$-convergence of tensor fields with respect to the Gromov-Hausdorff topology and several fundamental properties of the convergence. We apply this to establish a Bochner-type inequality which keeps the term of Hessian on the Gromov-Hausdorff limit space of a sequence of Riemannian manifolds with a lower Ricci curvature bound and to give a geometric explicit formula for the Dirichlet Laplacian on a limit space defined by Cheeger-Colding. We also prove a continuity of the first eigenvalues of the p-Laplacian with respect to the Gromov-Hausdorff topology.