Related papers: Isoperimetric inequality in noncompact MCP spaces

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 study minimal hypersurfaces in manifolds of non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay at infinity. By comparison with capped spherical cones, we identify a precise borderline for the Ricci…

In this paper, we study deeply geometric and topological properties of Finsler metric measure manifolds with the integral weighted Ricci curvature bounds. We first establish Laplacian comparison theorem, Bishop-Gromov type volume comparison…

We prove that, given an $RCD^{*}(K,N)$-space $(X,d,m)$, then it is possible to $m$-essentially cover $X$ by measurable subsets $(R_{i})_{i\in \mathbb{N}}$ with the following property: for each $i$ there exists $k_{i} \in \mathbb{N}\cap…

We study the qualitative stability of two classes of Sobolev inequalities on Riemannian manifolds. In the case of positive Ricci curvature, we prove that an almost extremal function for the sharp Sobolev inequality is close to an extremal…

Let $(M^n,g)$ be a complete Riemannian manifold which is not isometric to $\mathbb{R}^n$, has nonnegative Ricci curvature, Euclidean volume growth, and quadratic Riemann curvature decay. We prove that there exists a set $\mathcal{G}\subset…

We provide an isoperimetric inequality for critical metrics of the volume functional with nonnegative scalar curvature on compact manifolds with boundary. In addition, we establish a Weitzenb\"ock type formula for critical metrics of the…

We consider capillarity functionals which measure the perimeter of sets contained in a Euclidean half-space assigning a constant weight $\lambda \in (-1,1)$ to the portion of the boundary that touches the boundary of the half-space.…

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…

In this work, we establish a local smoothing result on metrics with small curvature concentration with respect to Sobolev constants and volume growth. In contrast with all previous works, we remove the Ricci curvature condition and…

In this expository paper, we discuss a unified framework for proving various geometric inequalities, based on the so-called Alexandrov-Bakelman-Pucci technique. Examples include Cabr\'e's proof of the classical isoperimetric inequality in…

We compare two approaches to Ricci curvature on non-smooth spaces, in the case of the discrete hypercube $\{0,1\}^N$. While the coarse Ricci curvature of the first author readily yields a positive value for curvature, the displacement…

The goal of the paper is four-fold. In the setting of non-smooth spaces with Ricci curvature lower bounds (more precisely RCD(K,N) metric measure spaces): - we develop an intrinsic theory of Laplacian bounds in viscosity sense and in a…

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…

In this article we study stability and compactness w.r.t. measured Gromov-Hausdorff convergence of smooth metric measure spaces with integral Ricci curvature bounds. More precisely, we prove that a sequence of $n$-dimensional Riemannian…

In an attempt to develop higher-dimensional quasiconformal mappings on metric measure spaces with curvature conditions, i.e. from Ahlfors to Alexsandrov, we show that a non-collapsed $\mathrm{RCD}(0,n)$ space ($n\geq2$) with Euclidean…

We establish a quantitative isoperimetric inequality for weighted Riemannian manifolds with $\mathrm{Ric}_{\infty} \ge 1$. Precisely, we give an upper bound of the volume of the symmetric difference between a Borel set and a sub-level (or…

We prove that an open manifold with nonnegative Ricci curvature, linear volume growth and noncollapsed ends always splits off a line at infinity. This completes the final step to prove the existence of isoperimetric sets given large volumes…

In this paper, we investigate the reverse improvement property of Sobolev inequalities on manifolds with quadratically decaying Ricci curvature. Specifically, we establish conditions under which the uniform decay of the heat kernel implies…

We study fine properties of the principal frequency of clamped plates in the (possibly singular) setting of metric measure spaces verifying the RCD(0,N) condition, i.e., infinitesimally Hilbertian spaces with non-negative Ricci curvature…