Nonexistence of isoperimetric sets in spaces of positive curvature
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866913367474569216 |
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| author | Antonelli, Gioacchino Glaudo, Federico |
| author_facet | Antonelli, Gioacchino Glaudo, Federico |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_13052 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Nonexistence of isoperimetric sets in spaces of positive curvature Antonelli, Gioacchino Glaudo, Federico Differential Geometry Metric Geometry 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. |
| title | Nonexistence of isoperimetric sets in spaces of positive curvature |
| topic | Differential Geometry Metric Geometry |
| url | https://arxiv.org/abs/2306.13052 |