The isoperimetric problem on Riemannian manifolds via Gromov-Hausdorff asymptotic analysis

In this paper we prove the existence of isoperimetric regions of any volume in Riemannian manifolds with Ricci bounded below assuming Gromov–Hausdorff asymptoticity to the suitable simply connected model 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 (possibly nonsmooth) 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. MSC (2020). Primary: 49J45, 26B30, 53A35. Secondary: 53C23, 49J52.

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