Related papers: Beyond traditional Curvature-Dimension I: new mode…

We obtain new sharp isoperimetric inequalities on a Riemannian manifold equipped with a probability measure, whose generalized Ricci curvature is bounded from below (possibly negatively), and generalized dimension and diameter of the convex…

It is well known that isoperimetric inequalities imply in a very general measure-metric-space setting appropriate concentration inequalities. The former bound the boundary measure of sets as a function of their measure, whereas the latter…

Let $\mathbb{M}$ be a smooth connected manifold endowed with a smooth measure $\mu$ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$, and which is symmetric with respect to $\mu$. We show that if $L$ satisfies, with…

We establish new sharp inequalities of Poincar\'{e} or log-Sobolev type, on geodesically-convex weighted Riemannian manifolds $(M,\mathfrak{g},\mu)$ whose (generalized) Ricci curvature $Ric_{\mathfrak{g},\mu,N}$ with effective dimension…

Aim of this paper is to provide new characterizations of the curvature dimension condition in the context of metric measure spaces (X,d,m). On the geometric side, our new approach takes into account suitable weighted action functionals…

We provide an isoperimetric comparison theorem for small volumes in an $n$-dimensional Riemannian manifold $(M^n,g)$ with strong bounded geometry, as in Definition $2.3$, involving the scalar curvature function. Namely in strong bounded…

We extend several Cheeger-type isoperimetric bounds for convex sets in Euclidean space, due to Bobkov and Kannan-Lov\'asz-Simonovits, to Riemannian manifolds having non-negative Ricci curvature. In order to extend Bobkov's bound, we require…

We study the space of Riemannian metrics with positive scalar curvature on a compact manifold with boundary. These metrics extend a fixed boundary metric and take a product structure on a collar neighbourhood of the boundary. We show that…

We prove that a Ricci curvature based method of triangulation of compact Riemannian manifolds, due to Grove and Petersen, extends to the context of weighted Riemannian manifolds and more general metric measure spaces. In both cases the role…

Positive curvature and bosons Compact positive curvature Riemannian manifolds M with symmetry group G allow Conner-Kobayashi reductions M to N, where N is the fixed point set of the symmetry G. The set N is a union of smaller-dimensional…

Let $({M},\textsf{d},\textsf{m})$ be a metric measure space which satisfies the Lott-Sturm-Villani curvature-dimension condition $\textsf{CD}(K,n)$ for some $K\geq 0$ and $n\geq 2$, and a lower $n-$density assumption at some point of $M$.…

The sharp isoperimetric inequality for non-compact Riemannian manifolds with non-negative Ricci curvature and Euclidean volume growth has been obtained in increasing generality with different approaches in a number of contributions…

We prove that if $(X,\mathsf{d},\mathfrak{m})$ is a metric measure space with $\mathfrak{m}(X)=1$ having (in a synthetic sense) Ricci curvature bounded from below by $K>0$ and dimension bounded above by $N\in [1,\infty)$, then the classic…

In this paper, I shall demonstrate that sufficiently high-dimensional closed positively-curved Riemannian manifolds are either diffeomorphic to a spherical space form, or isometric to a locally compact rank one symmetric space. This…

In this paper, we are concerned with noncollapsed Riemannian manifolds $(M^{n},g)$ with integral curvature bounds, as well as their Gromov-Hausdorff limits $(M^{n}_{i},g_{i})\xrightarrow{GH}(X,d)$. Our main result generalizes Cheeger's…

We prove that the isoperimetric profile of a convex domain $\Omega$ with compact closure in a Riemannian manifold $(M^{n+1},g)$ satisfies a second order differential inequality which only depends on the dimension of the manifold and on a…

Extending the celebrated results of Alexandrov (1958) and Korevaar-Ros (1988) for smooth sets, as well as the results of Schneider (1979) and the first author (1999) for arbitrary convex bodies, we obtain for the first time the…

If a graph submanifold $(x,f(x))$ of a Riemannian warped product space $(M^m\times_{e^{\psi}}N^n,\tilde{g}=g+e^{2\psi}h)$ is immersed with parallel mean curvature $H$, then we obtain a Heinz type estimation of the mean curvature. Namely, on…

By using optimal mass transport theory we prove a sharp isoperimetric inequality in ${\sf CD} (0,N)$ metric measure spaces assuming an asymptotic volume growth at infinity. Our result extends recently proven isoperimetric inequalities for…

In this paper we study sectional curvature bounds for Riemannian manifolds with density from the perspective of a weighted torsion free connection introduced recently by the last two authors. We develop two new tools for studying weighted…