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

created by ambrosio on 13 May 2016
modified on 14 Jun 2016

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Submitted Paper

Inserted: 13 may 2016
Last Updated: 14 jun 2016

Year: 2016

Abstract:

The aim of this paper is to provide new stability results for sequences of metric measure spaces $(X_i,\dist_i,\meas_i)$ convergent in the measured Gromov-Hausdorff sense. By adopting the so-called extrinsic approach of embedding all metric spaces into a common one $(X,\dist)$, we extend the results of \cite{GigliMondinoSavare13} 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 \cite{AmbrosioStraTrevisan}, 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 in \cite{Ketterer15a,Ketterer15b,CavallettiMondino15a,CavallettiMondino15b}. On the basis of the second-order calculus in \cite{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.