Calculus of Variations and Geometric Measure Theory

N. Gigli - A. Mondino - G. Savaré

Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows

created by mondino on 19 Nov 2013
modified on 24 Mar 2016


Accepted Paper

Inserted: 19 nov 2013
Last Updated: 24 mar 2016

Journal: Proceedings of the London Mathematical Society
Year: 2013
Doi: DOI: 10.1112/plms/pdv047


Aim of this paper is to discuss convergence of pointed metric measure spaces in absence of any compactness condition. We propose various definitions, show that all of them are equivalent and that for doubling spaces these are also equivalent to the well known measured-Gromov-Hausdorff convergence. Then we show that the curvature conditions $CD(K,\infty)$ and $RCD(K,\infty)$ are stable under this notion of convergence and that the heat flow passes to the limit as well, both in the Wasserstein and in the $L^2$-framework. We also prove the variational convergence of Cheeger energies in the naturally adapted $\Gamma$-Mosco sense and the convergence of the spectra of the Laplacian in the case of spaces either uniformly bounded or satisfying the $RCD(K,\infty)$ condition with $K>0$. When applied to Riemannian manifolds, our results allow for sequences with diverging dimensions.