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

Using techniques of optimal transportation and gradient flows in metric spaces, we extend the notion of Riemannian Curvature Dimension condition $RCD(K,\infty)$ introduced (in case the reference measure is finite) by Giuseppe Savare', the first and the second author, to the case the reference measure is $\sigma$-finite; in this way the theory includes natural examples as the euclidean $n$-dimensional space endowed with the Lebesgue measure, and noncompact manifolds with bounded geometry endowed with the Riemannian volume measure. Another major goal of the paper is to simplify the axiomatization of $RCD(K,\infty)$ (even in case of finite reference measure) replacing the assumption of strict $CD(K,\infty)$ with the classic notion of $CD(K,\infty)$.Comment: 42 pages; final version (minor changes to the old one, in particular we added some more preliminaries and explanations) to be published in Transactions of the AM

Perimeter as relaxed Minkowski content in metric measure spaces

In this note we prove that on general metric measure spaces the perimeter is equal to the relaxation of the Minkowski content w.r.t.\ convergence in measur

Metric measure spaces with Riemannian Ricci curvature bounded from below

In this paper we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces (X,d,m) which is stable under measured Gromov-Hausdorff convergence and rules out Finsler geometries. It can be given in terms of an enforcement of the Lott, Sturm and Villani geodesic convexity condition for the entropy coupled with the linearity of the heat flow. Besides stability, it enjoys the same tensorization, global-to-local and local-to-global properties. In these spaces, that we call RCD(K,\infty) spaces, we prove that the heat flow (which can be equivalently characterized either as the flow associated to the Dirichlet form, or as the Wasserstein gradient flow of the entropy) satisfies Wasserstein contraction estimates and several regularity properties, in particular Bakry-Emery estimates and the L^\infty-Lip Feller regularization. We also prove that the distance induced by the Dirichlet form coincides with d, that the local energy measure has density given by the square of Cheeger's relaxed slope and, as a consequence, that the underlying Brownian motion has continuous paths. All these results are obtained independently of Poincar\'e and doubling assumptions on the metric measure structure and therefore apply also to spaces which are not locally compact, as the infinite-dimensional onesComment: (v2) Minor typos, proof of Proposition 2.3, proof of Theorem 4.8: corrected. Proof of Theorem 6.2: corrected and simplified, thanks to the new Lemma 2.8. Lemma 3.6 and 4.6 (of v1) removed, since no more neede

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, 1e) and RCD(K, 1e) 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 L2-framework. We also prove the variational convergence of Cheeger energies in the naturally adapted \u393-Mosco sense and the convergence of the spectra of the Laplacian in the case of spaces either uniformly bounded or satisfying the RCD(K, 1e) condition with K>0. When applied to Riemannian manifolds, our results allow for sequences with diverging dimensions

A strong form of the Quantitative Isoperimetric inequality

We give a refinement of the quantitative isoperimetric inequality. We prove that the isoperimetric gap controls not only the Fraenkel asymmetry but also the oscillation of the boundary

Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below

This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces (X,d,m). Our main results are: - A general study of the relations between the Hopf-Lax semigroup and Hamilton-Jacobi equation in metric spaces (X,d). - The equivalence of the heat flow in L^2(X,m) generated by a suitable Dirichlet energy and the Wasserstein gradient flow of the relative entropy functional in the space of probability measures P(X). - The proof of density in energy of Lipschitz functions in the Sobolev space W^{1,2}(X,d,m). - A fine and very general analysis of the differentiability properties of a large class of Kantorovich potentials, in connection with the optimal transport problem. Our results apply in particular to spaces satisfying Ricci curvature bounds in the sense of Lott & Villani [30] and Sturm [39,40], and require neither the doubling property nor the validity of the local Poincar\'e inequality.Comment: Minor typos corrected and many small improvements added. Lemma 2.4, Lemma 2.10, Prop. 5.7, Rem. 5.8, Thm. 6.3 added. Rem. 4.7, Prop. 4.8, Prop. 4.15 and Thm 4.16 augmented/reenforced. Proof of Thm. 4.16 and Lemma 9.6 simplified. Thm. 8.6 corrected. A simpler axiomatization of weak gradients, still equivalent to all other ones, has been propose

The Abresch-Gromoll inequality in a non-smooth setting

We prove that the Abresch-Gromoll inequality holds on infinitesimally Hilbertian CD(K,N) spaces in the same form as the one available on smooth Riemannian manifolds

Characterization of the human STAT5A and STAT5B promoters: evidence of a positive and negative mechanism of transcriptional regulation

AbstractWe recently published the genomic characterization of the STAT5A and STAT5B paralogous genes that are located head to head in the 17q21 chromosome and share large regions of sequence identity. We here demonstrate by transient in vitro transfection that STAT5A and STAT5B promoters are able to direct comparable levels of transcription. The expression of basal promoters is enhanced after Sp1 up-regulation in HeLa and SL2 cells while DNA methylation associated to the recruitment of MeCP2 methyl CpG binding protein down-regulates STAT5A and B promoters by interfering with Sp1-induced transcription. In addition, cross-species sequence comparison identified a bi-directional negative cis-acting regulatory element located in the STAT5 intergenic region

Anti-angiogenic activity evaluation of secondary metabolites from Calycolpus moritzianus leaves.

Angiogenesis is a crucial step in many pathological conditions like cancer, inflammation and metastasis formation; on these basis the search for antiangiogenic agents has widened. In order to identify new compounds able to interfere in the Vascular Endothelial Growth Factor Receptor-1 (VEGFR-1, also known as Flt-1) recognition by VEGFs family members, we screened Calycolpus moritzianus (O. Berg) Burret leaves extracts by a competitive ELISA-based assay. MeOH and CHCl3 extracts and several their fractions demonstrated to be able to prevent VEGF or PlGF interaction with Flt-1, with an inhibition about 50% at concentration of 100 μg/mL. Phytochemical and pharmacological investigation of the active fractions led to the isolation of flavonoids, and terpenes