Related papers: Sharp isoperimetric comparison on non-collapsed sp…

We propose a definition of non-collapsed space with Ricci curvature bounded from below and we prove the versions of Colding's volume convergence theorem and of Cheeger-Colding dimension gap estimate for ${\sf RCD}$ spaces. In particular…

The Euclidean mixed isoperimetric-isodiametric inequality states that the round ball maximizes the volume under constraint on the product between boundary area and radius. The goal of the paper is to investigate such mixed…

We prove that $n$-dimensional ($n\geqslant3$) complete and non-compact metric measure spaces with non-negative weighted Ricci curvature in which some Caffarelli-Kohn-Nirenberg type inequality holds are close to the model metric measure…

We prove a lower bound for the first Steklov eigenvalue of embedded minimal hypersurfaces with free boundary in a compact $n$-dimensional manifold which has nonnegative Ricci curvature and strictly convex boundary. When $n=3$, this implies…

Klartag's needle decomposition technique enables one to obtain strong isoperimetric inequalities on Riemannian manifolds other than the classical known examples. As a result, in this paper, we obtain sharp isoperimetric inequalities for…

We show that the existence of an embedded compact, boundaryless hypersurface S of strictly positive mean curvature in a noncompact, connected, complete Riemannian n-manifold N of nonnegative Ricci curvature implies that the homomorphism…

We study the cases of equality and prove a rigidity theorem concerning the 1-Bakry-\'Emery inequality. As an application, we prove the rigidity of the Gaussian isoperimetric inequality, the logarithmic Sobolev inequality and the Poincar\'e…

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 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…

We consider the problem of minimizing the relative perimeter under a volume constraint in an unbounded convex body $C\subset \mathbb{R}^{n+1}$, without assuming any further regularity on the boundary of $C$. Motivated by an example of an…

We study comparison estimates on metric measure spaces admitting a synthetic variable Ricci curvature lower bound. We obtain geometric and functional inequalities assuming that the deficit of the lower bound from a given constant is…

We prove that a proper geodesic metric space has non-positive curvature in the sense of Alexandrov if and only if it satisfies the Euclidean isoperimetric inequality for curves. Our result extends to non-geodesic spaces and non-zero…

Let $(X_i,p_i)$ be a non-collapsing sequence of pointed $n$-dimensional Riemannian manifolds with a uniform lower Ricci curvature bound, and $G_i \leq \text{Iso} (X_i)$ a sequence of closed subgroups of isometries. We show that if the…

We prove a new kind of estimate that holds on any manifold with lower Ricci bounds. It relates the geometry of two small balls with the same radius, potentially far apart, but centered in the interior of a common minimizing geodesic. It…

We obtain new topological restrictions for complete Riemannian manifolds with nonnegative Ricci curvature and RCD(0,n) spaces. Our main results are a Betti number rigidity theorem which answers a question open since work of M.-T. Anderson…

We prove topological regularity results for isoperimetric sets in PI spaces having a suitable deformation property, which prescribes a control on the increment of the perimeter of sets under perturbations with balls. More precisely, we…

We study Riemannian manifolds $(M^n,g)$ with mean-convex boundary whose Ricci curvature is nonnegative in a spectral sense. Our first main result is a sharp spectral extension of a rigidity theorem by Kasue: we prove that under the…

In this paper we study spaces of Riemannian metrics with lower bounds on intermediate curvatures. We show that the spaces of metrics of positive p-curvature and k-positive Ricci curvature on a given high-dimensional Spin-manifold have many…

We study the isoperimetric, functional and concentration properties of $n$-dimensional weighted Riemannian manifolds satisfying the Curvature-Dimension condition, when the generalized dimension $N$ is negative, and more generally, is in the…

We investigate the topological regularity and stability of noncollapsed Ricci limit spaces $(M_i^n,g_i,p_i)\to (X^n,d)$. We confirm a conjecture proposed by Colding and Naber in dimension $n=4$, showing that the cross-sections of tangent…