Sharp isoperimetric comparison on non collapsed spaces with lower Ricci bounds

Abstract

This paper studies sharp and rigid isoperimetric comparison theorems and sharp dimensional concavity properties of the isoperimetric profile for non smooth spaces with lower Ricci curvature bounds, the so-called N -dimensional RCD(K, N ) spaces (X, d, H N ). Thanks to these results, we determine the asymptotic isoperimetric behaviour for small volumes in great generality, and for large volumes when K = 0 under an additional noncollapsing assumption. Moreover, we obtain new stability results for isoperimetric regions along sequences of spaces with uniform lower Ricci curvature and lower volume bounds, almost regularity theorems formulated in terms of the isoperimetric profile, and enhanced consequences at the level of several functional inequalities.The absence of most of the classical tools of Geometric Measure Theory and the possible non existence of isoperimetric regions on non compact spaces are handled via an original argument to estimate first and second variation of the area for isoperimetric sets, avoiding any regularity theory, in combination with an asymptotic mass decomposition result of perimeter-minimizing sequences.Most of our statements are new even for smooth, non compact manifolds with lower Ricci curvature bounds and for Alexandrov spaces with lower sectional curvature bounds. They generalize several results known for compact manifolds, non compact manifolds with uniformly bounded geometry at infinity, and Euclidean convex bodies. Contents 1. Introduction Isoperimetry and lower Ricci curvature bounds Main results Consequences and strategies of the proofs Sharp asymptotic behaviour for small and large volumes Comparison with the previous literature Acknowledgements 2. Preliminaries 2.1. Convergence and stability results 2.2. BV functions and sets of finite perimeter in metric measure spaces 2.3. Sobolev functions, Laplacians and vector fields in metric measure spaces 2.4. Geometric Analysis on RCD spaces 2.5. Localization of the Curvature-Dimension condition 3. The distance function from isoperimetric sets 4. Concavity properties of the isoperimetric profile function and consequences 4.1. Sharp concavity inequalities for the isoperimetric profile 4.2. Fine properties of the isoperimetric profile 4.3. Consequences 5. Stability of isoperimetric sets

The isoperimetric problemviadirect method in noncompact metric measure spaces with lower Ricci bounds

“…This result gives a mild regularity of the isoperimetric profile that will be sufficient for the purposes of this work. We stress that, by using refined tools of geometric analysis in nonsmooth spaces, and building on the regularity result of Lemma 2.23, much more can be said on the isoperimetric profile function and we address this problem in [17]. On the other hand the proof of the next result, which is adapted from [56,Theorem 2], is based on elementary comparison arguments.…”

“…In [17], the above results are crucially exploited to show useful properties of the isoperimetric profile of RCD(K, N ) spaces, without assumptions on the existence of isoperimetric sets. Building on such properties we shall prove a more explicit upper bound on the number N in Theorem 1.1, which turns out to be bounded from above linearly in terms of the volume V .…”

“…For instance, it follows from Lemma 5.3 that if I X is strictly subadditive on (0, V ] and I ∞ X (V ) > I X (V ), then item (1) is satisfied, and thus by Theorem 1.4 existence of isoperimetric regions of volume V holds. After [17], this reasoning applies, for example, to every noncompact RCD(0, N ) space (X, d, H N ) (different from R N ) that is Gromov-Hausdorff asymptotic to R N at infinity. This also extends to the nonsmooth setting some existence theorems in [15].…”

“…Moreover, the existence of isoperimetric regions has been crucial for the derivation of differential properties of the isoperimetric profile [19,53,21,20,22]. As shown in [17], Theorem 1.1 is a crucial ingredient for proving such properties without assuming existence of isoperimetric regions, as well as, for deriving isoperimetric inequalities and other geometric functional inequalities like the ones contained in [1,18,24].…”

On the existence of isoperimetric regions in manifolds with nonnegative Ricci curvature and Euclidean volume growth

“…The first and the fourth author very recently provided, in a joint work with E. Pasqualetto and D. Semola[15], variational tools capable of estimating the second variation of the perimeter in nonsmooth spaces, giving an alternative proof of the concavity properties of the isoperimetric profile in our setting.…”