New stability results for sequences of metric measure spaces with uniform Ricci bounds from below

TL;DR

This paper establishes new stability results for sequences of metric measure spaces with Ricci curvature bounds, extending previous work to include Sobolev spaces, spectral properties, and second-order calculus in the measured Gromov-Hausdorff convergence framework.

Contribution

It extends stability results to Sobolev spaces, spectral invariants, and second-order calculus in metric measure spaces with Ricci bounds, including variable exponents and local convergence of derivations.

Findings

Mosco convergence of Cheeger's energies in all Sobolev spaces

Continuity of Cheeger's constant under convergence

Stability of Hessians and $W^{2,2}$ functions in $RCD(K, olinebreak ext{infinity})$ spaces

Abstract

The aim of this paper is to provide new stability results for sequences of metric measure spaces $(X_i,d_i,m_i)$ convergent in the measured Gromov-Hausdorff sense. By adopting the so-called extrinsic approach of embedding all metric spaces into a common one $(X,d)$, we extend the results of Gigli-Mondino-Savar\'e by providing Mosco convergence of Cheeger's energies and compactness theorems in the whole range of Sobolev spaces $H^{1,p}$, including the space $BV$, and even with a variable exponent $p_i\in [1,\infty]$. In addition, building on the results of Ambrosio-Stra-Trevisan, we provide local convergence results for gradient derivations. We use these tools to improve the spectral stability results, previously known for $p>1$ and for Ricci limit spaces, getting continuity of Cheeger's constant. In the dimensional case $N<\infty$, we improve some rigidity and almost rigidity results by Ketterer and Cavaletti-Mondino. On the basis of the second-order calculus by Gigli, in the class of $RCD(K,\infty)$ spaces we provide stability results for Hessians and $W^{2,2}$ functions and we treat the stability of the Bakry-\'Emery condition $BE(K,N)$ and of ${\bf Ric}\geq KI$, with $K$ and $N$ not necessarily constant.