Nonexistence of isoperimetric sets in spaces of positive curvature

TL;DR

This paper constructs examples of smooth, positively curved, noncompact manifolds and convex sets in Euclidean space where isoperimetric sets do not exist for small volumes, highlighting limitations in isoperimetric problem solutions.

Contribution

It provides the first known examples of noncollapsed, nonnegatively curved spaces lacking isoperimetric sets for small volumes, with sharp dimensional constraints.

Findings

Constructed noncompact manifolds with positive curvature lacking small-volume isoperimetric sets.

Provided examples in convex sets in Euclidean space with similar properties.

Showed that for large volumes, isoperimetric sets always exist in these spaces.

Abstract

For every $d\ge 3$, 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 $\mathbb R^d$. The examples we construct have nondegenerate asymptotic cone. The dimensional constraint $d\ge 3$ 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.