Asymptotic isoperimetry on non collapsed spaces with lower Ricci bounds

Abstract

This paper studies sharp and rigid isoperimetric comparison theorems and asymptotic isoperimetric properties for small and large volumes on $N$-dimensional $\mathrm{RCD}(K,N)$ spaces $(X,\mathsf{d},{\mathcal{H}}^N)$. Moreover, we obtain almost regularity theorems formulated in terms of the isoperimetric profile and enhanced consequences at the level of several functional inequalities. Most of our statements are new even in the classical setting of smooth, non compact manifolds with lower Ricci curvature bounds. The synthetic theory plays a key role via compactness and stability arguments.

arXiv commentThis is the second of two companion papers originally appeared in a joint version in arXiv:2201.04916v1. The first of the two companion papers is arXiv:2201.04916

References in corpus (17)

• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below
• Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces
• Structure Theory of Metric-Measure Spaces with Lower Ricci Curvature Bounds
• Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds
• On the topology and the boundary of N-dimensional RCD(K,N) spaces
• New formulas for the Laplacian of distance functions and applications
• Asymptotic isoperimetry on non collapsed spaces with lower Ricci bounds
• On the existence of isoperimetric regions in manifolds with nonnegative Ricci curvature and Euclidean volume growth
• Isoperimetric sets in spaces with lower bounds on the Ricci curvature
• Polya-Szego inequality and Dirichlet $p$-spectral gap for non-smooth spaces with Ricci curvature bounded below
• Existence of isoperimetric regions in non-compact Riemannian manifolds under Ricci or scalar curvature conditions
• Isoperimetric structure of asymptotically conical manifolds
• Local Hölder continuity of the isoperimetric profile in complete noncompact Riemannian manifolds with bounded geometry
• Almost euclidean Isoperimetric Inequalities in spaces satisfying local Ricci curvature lower bounds
• The isoperimetric problem via direct method in noncompact metric measure spaces with lower Ricci bounds
• Sharp isoperimetric inequalities for small volumes in complete noncompact Riemannian manifolds of bounded geometry involving the scalar curvature

Cited by in corpus (7)

• Asymptotic isoperimetry on non collapsed spaces with lower Ricci bounds
• Sharp log-Sobolev inequalities in $\mathsf{CD}(0,N)$ spaces with applications
• Topological regularity of isoperimetric sets in PI spaces having a deformation property
• Isoperimetric problem and structure at infinity on Alexandrov spaces with nonnegative curvature
• Optimal Transport Approach to Michael-Simon-Sobolev Inequalities in Manifolds with Intermediate Ricci Curvature Lower Bounds
• Principal frequency of clamped plates on RCD(0,N) spaces: sharpness, rigidity and stability
• From bubbles to clusters: Multiple solutions to the Allen--Cahn system