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 RCD(K, N ) spaces (X, d, 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.

Isoperimetric problem and structure at infinity on Alexandrov spaces with nonnegative curvature

“…We recall the forthcoming result, which has been proved in [22,Proposition 3.1], see also [23,Proposition 3.11] for the last part of the statement. Theorem 2.20 ([23, 22]).…”

“…We finally recall some basic properties of the isoperimetric profile and the asymptotic mass decomposition on N -dimensional RCD(0, N ) spaces. The following Theorem 2.24 is obtained in [22,Theorem 3.8], specializing the study on the differential properties of the isoperimetric profile in the case of nonnegative curvature. The result in Proposition 2.…”

“…In the last fifty years, a great interest has been devoted to the study of several aspects of the isoperimetric problem on spaces with (classical or synthetic) notions of lower bounds on the curvature. We mention [32,31,27,59,89,83,84,41] concerning sharp, rigid and quantitative isoperimetric inequalities on compact spaces with Ricci lower bounds, [1,34,26,63,22,42,43] about sharp and rigid isoperimetric inequalities on noncompact spaces with nonnegative Ricci curvature and Euclidean volume growth, see (1.1) below, [27,113,89,73,28,29,30,86,23] for what concerns differential properties of the isoperimetric profile of spaces with Ricci lower bounds, and [102,104,90,91,86,49,19,18,20] studying existence of isoperimetric sets on noncompact spaces with lower curvature bounds. Specializing to the case of convex bodies in the Euclidean space, a number of results about existence of minimizers, isoperimetric inequalities, stability and topology of isoperimetric sets have been obtained in [77,113,106,74].…”

“…Under a suitable assumption on the geometry at infinity of the space, in [18,Theorem 1.1] it is established existence of isoperimetric sets for any sufficiently large volume. Furthermore, if (X, d, H N ) is CBB(0) and AVR(X) > 0, then isoperimetric sets exist for any sufficiently large volume without further assumptions [18,Theorem 1.3], and [22,Theorem 1.2]. The results in [18] are generalized in the RCD setting in [22], see, e.g., [22,Theorem 1.2].…”

“…Furthermore, if (X, d, H N ) is CBB(0) and AVR(X) > 0, then isoperimetric sets exist for any sufficiently large volume without further assumptions [18,Theorem 1.3], and [22,Theorem 1.2]. The results in [18] are generalized in the RCD setting in [22], see, e.g., [22,Theorem 1.2].…”