Nonexistence of isoperimetric sets in spaces of positive curvature

Abstract

Abstract For every d ≥ 3 d\geq 3 , we construct a noncompact smooth 𝑑-dimensional Riemannian manifold with strictly positive sectional curvature without isoperimetric sets for any volume below 1. We construct a similar example also for the relative isoperimetric problem in (unbounded) convex sets in R d \mathbb{R}^{d} . The examples we construct have nondegenerate asymptotic cone. The dimensional constraint d ≥ 3 d\geq 3 is sharp. Our examples exhibit nonexistence of isoperimetric sets only for small volumes; indeed, in nonnegatively curved spaces with nondegenerate asymptotic cones, isoperimetric sets with large volumes always exist. This is the first instance of noncollapsed nonnegatively curved space without isoperimetric sets.