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

Luigi Ambrosio; Shouhei Honda

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arXiv · 1605.07908

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

Luigi Ambrosio·Shouhei Honda

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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-Savaré 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-Émery condition $BE(K,N)$ and of ${\bf Ric}\geq KI$, with $K$ and $N$ not necessarily constant.

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BibTeX RIS

Luigi Ambrosio, Shouhei Honda. 2016-07-05. New stability results for sequences of metric measure spaces with uniform Ricci bounds from below. https://arxiv.org/abs/1605.07908

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