Andrea Mondino

Riemannian Ricci curvature lower bounds in metric measure spaces with -finite measure

In prior work (4) of the first two authors with Savare, a new Riemannian notion of lower bound for Ricci curvature in the class of metric measure spaces (X,d,m) was introduced, and the corresponding class of spaces denoted by RCD(K,∞). This notion relates the CD(K,N) theory of Sturm and Lott-Villani, in the case N = ∞, to the Bakry-Emery approach. In (4) the RCD(K,∞) property is defined in three equivalent ways and several properties of RCD(K,∞) spaces, including the regularization properties of the heat flow, the connections with the theory of Dirichlet forms and the stability under tensor products, are provided. In (4) only finite reference measures m have been considered. The goal of this paper is twofold: on one side we extend these results to general σ-finite spaces, on the other we remove a technical assumption appeared in (4) concerning a strengthening of the CD(K,∞) condition. This more general class of spaces includes Euclidean spaces endowed with Lebesgue measure, complete noncompact Riemannian manifolds with bounded geometry and the pointed metric measure limits of manifolds with lower Ricci curvature bounds.

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

Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds

CD(K,\infty)

Li–Yau and Harnack type inequalities in RCD∗(K,N) metric measure spaces

and

Euclidean spaces as weak tangents of infinitesimally Hilbertian metric measure spaces with Ricci curvature bounded below

RCD(K,\infty)

On the volume measure of non-smooth spaces with Ricci curvature bounded below

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

Optimal maps in essentially non-branching spaces

L^2

Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds

-framework. We also prove the variational convergence of Cheeger energies in the naturally adapted

The Conformal Willmore Functional: A Perturbative Approach

\Gamma

Immersed Spheres of Finite Total Curvature into Manifolds.

-Mosco sense and the convergence of the spectra of the Laplacian in the case of spaces either uniformly bounded or satisfying the