Ricci curvature and convergence of Lipschitz functions

TL;DR

This paper introduces a new way to analyze the convergence of Lipschitz functions' differentials in measured Gromov-Hausdorff topology, with applications to harmonic functions and Laplacian comparison on limit spaces.

Contribution

It provides a novel definition of differential convergence for Lipschitz functions and applies it to characterize harmonic functions and establish Laplacian comparison theorems on limit spaces.

Findings

Characterization of harmonic functions with polynomial growth on asymptotic cones.

Distributional Laplacian comparison theorem on limit spaces.

New framework for convergence of differentials in measured Gromov-Hausdorff topology.

Abstract

We give a definition of convergence of differential of Lipschitz functions with respect to measured Gromov-Hausdorff topology. As their applications, we give a characterization of harmonic functions with polynomial growth on asymptotic cones of manifolds with nonnegative Ricci curvature and Euclidean volume growth, and distributional Laplacian comparison theorem on limit spaces of Riemannian manifolds.