The quadruple-bubble conjecture in three-dimensional space
For volumes $v_1, \dots, v_4 > 0$, consider clusters of pairwise disjoint sets of finite perimeter $E_1, \dots, E_4 \subset \mathbb{R}^3$ with $|E_i| = v_i$; the cells need not be connected. With $E_5 = \mathbb{R}^3 \setminus \bigcup_{i \le 4} E_i$, the total perimeter is $$P(E_1, \dots, E_4) = \frac12 \sum_{i=1}^{5} \operatorname{Per}(E_i),$$ so that each interface is counted once. A standard quadruple bubble is obtained as follows: choose unit vectors $c_1, \dots, c_5 \in \mathbb{R}^4$ with $c_i \cdot c_j = -1/4$ for $i \ne j$, partition $S^3$ into the cells $C_i = \{x \in S^3 : x \cdot c_i \ge x \cdot c_j \text{ for all } j\}$, project stereographically to $\mathbb{R}^3$ from a point in the interior of $C_5$, and apply a similarity. Sullivan conjectured that standard $k$-bubbles in $\mathbb{R}^n$ minimize total perimeter for their volumes whenever $k \le n + 1$; the case here is $k = 4$, $n = 3$. Hutchings, Morgan, Ritoré and Ros proved the double-bubble case in $\mathbb{R}^3$ (2002). Milman and Neeman proved that standard $k$-bubbles are the unique minimizers in $\mathbb{R}^n$ for $k \le \min(4, n)$, which includes the triple bubble in $\mathbb{R}^3$ and the quadruple bubble in $\mathbb{R}^n$ for $n \ge 4$ (2025).
Problem. Is every cluster minimizing $P$ among clusters with volumes $v_1, \dots, v_4$, up to null sets and Euclidean isometries, a standard quadruple bubble with these volumes?
Source: Sullivan, Problem 2 in Sullivan and Morgan (1996), for standard k-bubbles in ℝⁿ with k ≤ n + 1; the case of four bubbles in ℝ³ is singled out in Milman (2026), §9, problem (1).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DG — Differential Geometry
- Secondary
- MG, OC
- Keywords
- AIM Open Applied Problems, isoperimetric clusters, soap bubbles
References
@article{sullivan1996,
author = {Sullivan, John M. and Morgan, Frank},
title = {Open problems in soap bubble geometry},
journal = {Internat. J. Math.},
volume = {7},
number = {6},
year = {1996},
pages = {833--842},
doi = {10.1142/S0129167X9600044X},
}
@incollection{milman2026,
author = {Milman, Emanuel},
title = {Multi-bubble isoperimetric problems},
booktitle = {Proceedings of the International Congress of Mathematicians 2026, Vol. 4},
publisher = {SIAM},
year = {2026},
pages = {678--697},
doi = {10.1137/25M1804224},
eprint = {2510.07078},
archivePrefix = {arXiv},
}
@article{hutchings2002,
author = {Hutchings, Michael and Morgan, Frank and Ritor{\'e}, Manuel and Ros, Antonio},
title = {Proof of the double bubble conjecture},
journal = {Ann. of Math. (2)},
volume = {155},
number = {2},
year = {2002},
pages = {459--489},
doi = {10.2307/3062123},
}
@article{milman2025,
author = {Milman, Emanuel and Neeman, Joe},
title = {The structure of isoperimetric bubbles on $\mathbb{R}^n$ and $\mathbb{S}^n$},
journal = {Acta Math.},
volume = {234},
number = {1},
year = {2025},
pages = {71--188},
doi = {10.4310/ACTA.2025.v234.n1.a2},
eprint = {2205.09102},
archivePrefix = {arXiv},
}
@misc{aim-spatial-quadruple-bubble,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The quadruple-bubble conjecture in three-dimensional space}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/236-spatial-quadruple-bubble.md},
note = {Entry 236 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00644, "The quadruple-bubble conjecture in three-dimensional space", clopen.solutions.