Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows
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.
The paper is available on the
cvgmt preprint server.