Calculus of Variations and Geometric Measure Theory
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L. Ambrosio - S. Honda

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


Submitted Paper

Inserted: 13 may 2016
Last Updated: 14 jun 2016

Year: 2016


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.


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