Stability of the heat flow under convergence in concentration and consequences

TL;DR

This paper extends stability results of Sobolev functions and differential operators to convergence in concentration, establishing new inequalities and convergence of heat flows under Ricci curvature bounds in metric measure spaces.

Contribution

It introduces a framework for stability under concentration convergence, including Mosco-convergence of Cheeger energies and stability of heat flows and differential objects.

Findings

Established a $ ext{Gamma}$-$ ext{limsup}$ inequality for Cheeger energy.

Proved convergence of heat flows under uniform ${ m CD}(K, ext{infty})$ conditions.

Developed tools for convergence of maps between metric measure spaces in concentration.

Abstract

We extend to the framework of convergence in concentration virtually all the results concerning stability of Sobolev functions and differential operators known to be in place under the stronger measured-Gromov-Hausdorff convergence. These include, in particular: i) A general $\Gamma$--$\varlimsup$ inequality for the Cheeger energy, ii) Convergence of the heat flow under a uniform ${\sf CD}(K,\infty)$ condition on the spaces. As we will show, building on ideas developed for mGH-convergence, out of this latter result we can obtain clean stability statements for differential, flows of vector fields, Hessian, eigenvalues of the Laplacian and related objects in presence of uniform lower Ricci bounds. At the technical level, one of the tools we develop to establish these results is the notion of convergence of maps between metric measure spaces converging in concentration. Previous results in the field concerned the stability of the ${\sf CD}$ (Funano-Shioya '13) and ${\sf RCD}$ (Ozawa-Yokota '19) conditions. The latter was obtained proving $\Gamma$-convergence for the Cheeger energies. This is not sufficient to pass to the limit in the heat flow; one of the byproducts of our work is the improvement of this to Mosco-convergence.