In this paper we prove the existence of isoperimetric regions of any volume in Riemannian manifolds with Ricci bounded below and with a mild assumption at infinity, that is Gromov-Hausdorff asymptoticity to simply connected models of constant sectional curvature.The previous result is a consequence of a general structure theorem for perimeterminimizing sequences of sets of fixed volume on noncollapsed Riemannian manifolds with a lower bound on the Ricci curvature. We show that, without assuming any further hypotheses on the asymptotic geometry, all the mass and the perimeter lost at infinity, if any, are recovered by at most countably many isoperimetric regions sitting in some Gromov-Hausdorff limits at infinity.The Gromov-Hausdorff asymptotic analysis conducted allows us to provide, in low dimensions, a result of nonexistence of isoperimetric regions in Cartan-Hadamard manifolds that are Gromov-Hausdorff asymptotic to the Euclidean space.While studying the isoperimetric problem in the smooth setting, the nonsmooth geometry naturally emerges, and thus our treatment combines techniques from both the theories.

In this paper we prove the one-dimensional Preiss' theorem in the first Heisenberg group H 1 . More precisely we show that a Radon measure φ on H 1 with positive and finite one-density with respect to the Koranyi distance is supported on a one-rectifiable set in the sense of Federer, i.e., it is supported on the countable union of the images of Lipschitz mapsThe previous theorem is a consequence of a Marstrand-Mattila type rectifiability criterion, which we prove in arbitrary Carnot groups for measures with tangent planes that admit a normal complementary subgroup. Namely, in this co-normal case, even if we a priori ask that the tangent planes at a point might rotate at different scales, a posteriori the measure has a unique tangent almost everywhere. Since every horizontal subgroup has a normal complement, our criterion applies in the particular case of one-dimensional horizontal subgroups.These results are the outcome of a detailed study of a new notion of rectifiability: we say that a Radon measure on a Carnot group is P h -rectifiable, for h ∈ N, if it has positive h-lower density and finite h-upper density almost everywhere, and, at almost every point, it admits as tangent measures only (multiple of) the Haar measure of a homogeneous subgroup of Hausdorff dimension h.We also prove several structure properties of P h -rectifiable measures. First, we compare P h -rectifiability with other notions of rectifiability previously known in the literature in the setting of Carnot groups and we realize that it is strictly weaker than them. Furthermore, we show that a P h -rectifiable measure has almost everywhere positive and finite h-density whenever the tangents admit at least one complementary subgroup. Finally, we prove that a P h -rectifiabile measure with complemented tangents is supported on the union of intrinsically differentiable graphs.

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

This paper deals with the theory of rectifiability in arbitrary Carnot groups, and in particular with the study of the notion of $$\mathscr {P}$$ P -rectifiable measure. First, we show that in arbitrary Carnot groups the natural infinitesimal definition of rectifiabile measure, i.e., the definition given in terms of the existence of flat tangent measures, is equivalent to the global definition given in terms of coverings with intrinsically differentiable graphs, i.e., graphs with flat Hausdorff tangents. In general we do not have the latter equivalence if we ask the covering to be made of intrinsically Lipschitz graphs. Second, we show a geometric area formula for the centered Hausdorff measure restricted to intrinsically differentiable graphs in arbitrary Carnot groups. The latter formula extends and strengthens other area formulae obtained in the literature in the context of Carnot groups. As an application, our analysis allows us to prove the intrinsic $$C^1$$ C 1 -rectifiability of almost all the preimages of a large class of Lipschitz functions between Carnot groups. In particular, from the latter result, we obtain that any geodesic sphere in a Carnot group equipped with an arbitrary left-invariant homogeneous distance is intrinsic $$C^1$$ C 1 -rectifiable.

In this paper we provide new existence results for isoperimetric sets of large volume in Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth. We find sufficient conditions for their existence in terms of the geometry at infinity of the manifold. As a byproduct we show that isoperimetric sets of big volume always exist on manifolds with nonnegative sectional curvature and Euclidean volume growth. Our method combines an asymptotic mass decomposition result for minimizing sequences, a sharp isoperimetric inequality on nonsmooth spaces, and the concavity property of the isoperimetric profile. The latter is new in the generality of noncollapsed manifolds with Ricci curvature bounded below.