For every d ≥ 3 d\geq 3, we construct a noncompact smooth -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.
Nonexistence of isoperimetric sets in spaces of positive curvature
Antonelli, Gioacchino
;Glaudo, Federico
2024
Abstract
For every d ≥ 3 d\geq 3, we construct a noncompact smooth -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.File | Dimensione | Formato | |
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