Standard bubbles (and other M\"obius-flat partitions) on model spaces are stable

Abstract

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 M\"obius 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.