Isoperimetric inequalities in non-compact spaces

The sharp isoperimetric inequality for Riemannian manifolds having non-negative Ricci curvature and Euclidean volume growth has been obtained in increasing generality in a number of contributions [1, 46, 19, 51], culminated by Balogh and Kristály [14]. This last results covers also the case of non-smooth metric-measure spaces verifying the non-negative Ricci curvature condition in the synthetic sense of Sturm and Lott--Villani (the so-called CD condition) and the proof exploits the Brunn--Minkowski inequality. In the first part of the present thesis, we generalize the isoperimetric inequality to the wider class of MCP(0,N) spaces having Euclidean volume growth. The inequality we prove is sharp and the constant in the lower bound is smaller than the constant found in [14]. Since the Brunn–Minkowski is not available in MCP spaces, our proof is a scaling limit of the localization approach. In the second part, we present a characterization of the isoperimetric sets in the same generality of [14], i.e., the synthetic CD setting. Namely, we prove that the equality in the isoperimetric inequality can be attained only by metric balls and, whenever this happens, the space is forced, in a measure-theoretic sense, to be a cone. As a corollary, in the setting of general RCD spaces, we derive rigidity for the metric structure, i.e., the space is a cone also in the metric sense, generalizing a result of Antonelli et al. [10]. The proof consists in a careful refinement of the scaling argument presented in first part. In the third part, we generalize the isoperimetric inequality to the family of irreversible Finsler manifold with non-negative Ricci curvature and Euclidean volume growth. Irreversible Finsler manifolds are not covered by the theory of metric-measure spaces, for the distance induced by the Finsler structure is not symmetric. We also prove a rigidity result analogous to the result we obtained in the synthetic CD setting, namely a rigidity for the isoperimetric set and for the space, in the measure-theoretic sense. The proof of the inequality is based on the Brunn–Minkowski inequality; the rigidity is proved using the argument we developed in the second part. As a by-product of the rigidity results in both the CD and Finsler setting, we deduce that optimizers in the anisotropic and weighted isoperimetric inequality for Euclidean cones are necessarily the Wulff shapes.