Minimising hulls, p-capacity and isoperimetric inequality on complete Riemannian manifolds

Abstract

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.

On the existence of isoperimetric regions in manifolds with nonnegative Ricci curvature and Euclidean volume growth

“…To this aim we employ the recent result [17,Theorem 1.1], showing that the isoperimetric constant on CD(0, n) spaces is, roughly speaking, realized by balls of infinite radius, and thus that it is explicitly related to the asymptotic volume ratio. This result generalizes earlier analogous inequalities in the smooth setting [1,22,44,53]. Notice that the pmGH limits at infinity of a noncollpased nonnegatively Ricci curved manifold are indeed CD(0, n) spaces.…”

“…Here, we provide a version of the sharp isoperimetric inequality in CD(0, n) spaces recently obtained in [17] (see also the earlier [1,22,44,53]). Namely, an application of [7] allows to obtain it in terms of the perimeter, in place of the Minkowski content.…”

Comparison results for solutions of Poisson equations with Robin boundary on complete Riemannian manifolds

“…The equality holds in (2.1) if and only if (M, g) is isometric to (R n (κ), g κ ) and Ω is isometric to a geodesic ball in R n (κ). For a complete noncompact n-dimensional Riemannian manifold (M, g) with nonnegative Ricci curvature and positive asymptotic volume growth, we also notice that the inequality (2.1) is proved by Agostiniani et al in [1,Theorem 1.8] for n = 3 and then extended to 3 ≤ n ≤ 7 by Fogagnolo and Mazzieri in [18]. Furthermore, (2.1) and its equality case still hold in CD(0, N) metric measure spaces based on the method of optimal mass transport by Balogh and Kristály in [6].…”

Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds

“…We face now face a necessary step for the proof of Theorem 3.8 starting with the following local isoperimetric inequality of Euclidean type to be used in conjunction with Polya-Szego inequality developed in the previous section. The proof relies on the Brunn-Minkowski inequality and it is mainly inspired by [26], where sharp global isoperimetric inequalities for CD(0, N ) spaces have been proved (see also [17] for a refinement and the previous [31] and [48] for the smooth case). It is worth to mention that a class of "almost-Euclidean" isoperimetric inequalities in essentially nonbranching CD-spaces, similar to the following ones, were proved in [36] via localization-technique.…”