Uniqueness on average of large isoperimetric sets in noncompact manifolds with nonnegative Ricci curvature
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916626413125632 |
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| author | Antonelli, Gioacchino Pozzetta, Marco Semola, Daniele |
| author_facet | Antonelli, Gioacchino Pozzetta, Marco Semola, Daniele |
| contents | Let $(M^n,g)$ be a complete Riemannian manifold which is not isometric to $\mathbb{R}^n$, has nonnegative Ricci curvature, Euclidean volume growth, and quadratic Riemann curvature decay. We prove that there exists a set $\mathcal{G}\subset (0,\infty)$ with density $1$ at infinity such that for every $V\in \mathcal{G}$ there is a unique isoperimetric set of volume $V$ in $M$; moreover, its boundary is strictly volume preserving stable.
The latter result cannot be improved to uniqueness or strict stability for every large volume. Indeed, we construct a complete Riemannian surface satisfying the previous assumptions and with the following additional property: there exist arbitrarily large and diverging intervals $I_n\subset (0,\infty)$ such that isoperimetric sets with volumes $V\in I_n$ exist, but they are neither unique nor do they have strictly volume preserving stable boundaries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_07509 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniqueness on average of large isoperimetric sets in noncompact manifolds with nonnegative Ricci curvature Antonelli, Gioacchino Pozzetta, Marco Semola, Daniele Differential Geometry Metric Geometry Let $(M^n,g)$ be a complete Riemannian manifold which is not isometric to $\mathbb{R}^n$, has nonnegative Ricci curvature, Euclidean volume growth, and quadratic Riemann curvature decay. We prove that there exists a set $\mathcal{G}\subset (0,\infty)$ with density $1$ at infinity such that for every $V\in \mathcal{G}$ there is a unique isoperimetric set of volume $V$ in $M$; moreover, its boundary is strictly volume preserving stable. The latter result cannot be improved to uniqueness or strict stability for every large volume. Indeed, we construct a complete Riemannian surface satisfying the previous assumptions and with the following additional property: there exist arbitrarily large and diverging intervals $I_n\subset (0,\infty)$ such that isoperimetric sets with volumes $V\in I_n$ exist, but they are neither unique nor do they have strictly volume preserving stable boundaries. |
| title | Uniqueness on average of large isoperimetric sets in noncompact manifolds with nonnegative Ricci curvature |
| topic | Differential Geometry Metric Geometry |
| url | https://arxiv.org/abs/2406.07509 |