Standard bubbles (and other Möbius-flat partitions) on model spaces are stable
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866910911915098112 |
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| author | Milman, Emanuel Xu, Botong |
| author_facet | Milman, Emanuel Xu, Botong |
| contents | We verify that for all $n \geq 3$ and $2 \leq k \leq n+1$, the standard $k$-bubble clusters, conjectured to be minimizing total perimeter in $\mathbb{R}^n$, $\mathbb{S}^n$ and $\mathbb{H}^n$, are stable -- an infinitesimal regular perturbation preserving volume to first order yields a non-negative second variation of area modulo the volume constraint. In fact, stability holds for all standard $\textit{partitions}$, in which several cells are allowed to have infinite volume. In the Gaussian setting, any partition in $\mathbb{G}^n$ ($n\geq 2$) obeying Plateau's laws and whose interfaces are all $\textit{flat}$, is stable. Our results apply to non-standard partitions as well - starting with any (regular) flat Voronoi partition in $\mathbb{S}^n$ and applying Möbius transformations and stereographic projections, the resulting partitions in $\mathbb{R}^n$, $\mathbb{S}^n$ and $\mathbb{H}^n$ are stable. Our proof relies on a new conjugated Brascamp-Lieb inequality on partitions with conformally flat umbilical boundary, and the construction of a good conformally flattening boundary potential. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_11185 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Standard bubbles (and other Möbius-flat partitions) on model spaces are stable Milman, Emanuel Xu, Botong Differential Geometry Functional Analysis We verify that for all $n \geq 3$ and $2 \leq k \leq n+1$, the standard $k$-bubble clusters, conjectured to be minimizing total perimeter in $\mathbb{R}^n$, $\mathbb{S}^n$ and $\mathbb{H}^n$, are stable -- an infinitesimal regular perturbation preserving volume to first order yields a non-negative second variation of area modulo the volume constraint. In fact, stability holds for all standard $\textit{partitions}$, in which several cells are allowed to have infinite volume. In the Gaussian setting, any partition in $\mathbb{G}^n$ ($n\geq 2$) obeying Plateau's laws and whose interfaces are all $\textit{flat}$, is stable. Our results apply to non-standard partitions as well - starting with any (regular) flat Voronoi partition in $\mathbb{S}^n$ and applying Möbius transformations and stereographic projections, the resulting partitions in $\mathbb{R}^n$, $\mathbb{S}^n$ and $\mathbb{H}^n$ are stable. Our proof relies on a new conjugated Brascamp-Lieb inequality on partitions with conformally flat umbilical boundary, and the construction of a good conformally flattening boundary potential. |
| title | Standard bubbles (and other Möbius-flat partitions) on model spaces are stable |
| topic | Differential Geometry Functional Analysis |
| url | https://arxiv.org/abs/2504.11185 |