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Standard bubbles (and other Möbius-flat partitions) on model spaces are stable

Emanuel Milman, Botong Xu

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 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.

Standard bubbles (and other Möbius-flat partitions) on model spaces are stable

Abstract

We verify that for all and , the standard -bubble clusters, conjectured to be minimizing total perimeter in , and , 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 , in which several cells are allowed to have infinite volume. In the Gaussian setting, any partition in () obeying Plateau's laws and whose interfaces are all , is stable. Our results apply to non-standard partitions as well - starting with any (regular) flat Voronoi partition in and applying Möbius transformations and stereographic projections, the resulting partitions in , and 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.

Paper Structure

This paper contains 26 sections, 41 theorems, 170 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Informal presentation of main results
  3. Stability and the Jacobi operator
  4. Conjugated Brascamp-Lieb inequality on partitions with conformally flat umbilical boundary
  5. Construction of a good conformally flattening potential
  6. Preliminaries
  7. Weighted divergence and mean-curvature
  8. Partitions and interfaces
  9. Regularity
  10. Stationarity
  11. Locally bounded curvature
  12. Stability
  13. Generalized spheres and quasi-centers
  14. Spherical Voronoi and standard partitions
  15. Divergence Theorems
  16. ...and 11 more sections

Key Result

Theorem 1.2

Let $\Omega$ be a stationary regular partition of $(M^n, g, \mu)$, $n\geq 2$, with locally-bounded curvature and umbilical boundary. Assume that $\Omega$ has conformally flat boundary, and let $V>0$ be a conformally flattening boundary potential. Let $f = (f_{ij})$ be a scalar-field of the form $f = Then:

Figures (8)

  • Figure 1: Stereographic projection of an equipartition of $\mathbb{S}^2$ into 4 Voronoi cells, yielding a standard triple-bubble in $\mathbb{R}^2$.
  • Figure 2: Left: a standard quadruple-bubble in $\mathbb{R}^3$. Right: a $3$D cross-section of a sextuple-bubble in $\mathbb{R}^5$. Both are conjectured to be minimizing total perimeter under volume constraint.
  • Figure 3: Standard $5$-partitions in $\mathbb{R}^3$ with $2$ (left), $3$ (middle) and $4$ (right) unbounded cells. All three are conjectured to be locally minimizing perimeter under volume constraint.
  • Figure 4: A non-standard 7-bubble in $\mathbb{R}^3$ with a cubical inner cell, often created by soap-bubble magicians, is stable.
  • Figure 5: Non-standard 5-bubbles in $\mathbb{R}^3$ with two smaller cells highlighted (top) and with equal-volume cells enclosing a tetrahedral inner cell (bottom), are stable.
  • ...and 3 more figures

Theorems & Definitions (100)

  • Definition 1.1: Conformally flat boundary
  • Theorem 1.2: Conjugated Brascamp-Lieb inequality on partitions with conformally flat umbilical boundary
  • Remark 1.3
  • Remark 1.4
  • Proposition 1.5
  • Corollary 1.6
  • proof
  • Corollary 1.7
  • Theorem 1.8
  • Corollary 1.10
  • ...and 90 more