In this work we establish a sharp geometric inequality for closed hypersurfaces in complete noncompact Riemannian manifolds with asymptotically nonnegative curvature using standard comparison methods in Riemannian Geometry. These methods have been applied in a recent work by X. Wang [1] to greatly simplify the proof of a Willmore-type inequality in complete noncompact Riemannian manifolds of nonnegative Ricci curvature, which was first proved by Agostiniani, Fagagnolo and Mazzieri [2].

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.

We provide monotonicity formulas for solutions to the p-Laplace equation defined in the exterior of a convex domain. A number of analytic and geometric consequences are derived, including the classical Minkowski inequality as well as new characterizations of rotationally symmetric solutions and domains. The proofs rely on the conformal splitting technique introduced by the second author in collaboration with V. Agostiniani.

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.

The notion of strictly outward minimising hull is investigated for open sets of finite perimeter sitting inside a complete noncompact Riemannian manifold. Under natural geometric assumptions on the ambient manifold, the strictly outward minimising hull Ω * of a set Ω is characterised as a maximal volume solution of the least area problem with obstacle, where the obstacle is the set itself. In the case where Ω has C 1,α -boundary, the area of ∂Ω * is recovered as the limit of the p-capacities of Ω, as p → 1 + . Finally, building on the existence of strictly outward minimising exhaustions, a sharp isoperimetric inequality is deduced on complete noncompact manifolds with nonnegative Ricci curvature, provided 3 ≤ n ≤ 7.