The Structure of Isoperimetric Bubbles on $\mathbb{R}^n$ and $\mathbb{S}^n$

TL;DR

This work advances the multi-bubble isoperimetric theory by establishing that for q ≤ min(5,n+1), the isoperimetric minimizers in both Euclidean and spherical spaces are standard bubbles. The authors prove that minimizers in this regime have spherical interfaces and admit a spherical Voronoi representation after stereographic projection, with cells corresponding to intersections of convex polyhedra in R^{n+1}. Central to the method is a tandem analysis of R^n and S^n via Möbius geometry and conformal Killing fields, which enables a detailed structural decomposition into Voronoi cells and leverages symmetry (S^0) to deduce connectedness and equatorial properties. The work develops a robust variational framework—via first and second variation, index-forms Q, Jacobi operators, and conformal boundary conditions—while circumventing the need for a Gaussian-type PDI, making intricate use of curvature bounds, projection techniques, and discrete topology to resolve classical Heppes-type questions. Overall, the paper provides a rigorous, geometrically enriched route to standard bubble minimality in the first nontrivial multi-bubble regimes and lays groundwork for extensions to higher q under further geometric conditions.

Abstract

The multi-bubble isoperimetric conjecture in $n$-dimensional Euclidean and spherical spaces from the 1990's asserts that standard bubbles uniquely minimize total perimeter among all $q-1$ bubbles enclosing prescribed volume, for any $q \leq n+2$. The double-bubble conjecture on $\mathbb{R}^3$ was confirmed in 2000 by Hutchings-Morgan-Ritoré-Ros, and is nowadays fully resolved for all $n \geq 2$. The double-bubble conjecture on $\mathbb{S}^2$ and triple-bubble conjecture on $\mathbb{R}^2$ have also been resolved, but all other cases are in general open. We confirm the conjecture on $\mathbb{R}^n$ and on $\mathbb{S}^n$ for all $q \leq \min(5,n+1)$, namely: the double-bubble conjectures for $n \geq 2$, the triple-bubble conjectures for $n \geq 3$ and the quadruple-bubble conjectures for $n \geq 4$. In fact, we show that for all $q \leq n+1$, a minimizing cluster necessarily has spherical interfaces, and after stereographic projection to $\mathbb{S}^n$, its cells are obtained as the Voronoi cells of $q$ affine-functions, or equivalently, as the intersection with $\mathbb{S}^n$ of convex polyhedra in $\mathbb{R}^{n+1}$. Moreover, the cells (including the unbounded one) are necessarily connected and intersect a common hyperplane of symmetry, resolving a conjecture of Heppes. We also show for all $q \leq n+1$ that a minimizer with non-empty interfaces between all pairs of cells is necessarily a standard bubble. The proof makes crucial use of considering $\mathbb{R}^n$ and $\mathbb{S}^n$ in tandem and of Möbius geometry and conformal Killing fields; it does not rely on establishing a PDI for the isoperimetric profile as in the Gaussian setting, which seems out of reach in the present one.

Figures (14)

• Figure 1: Left: a standard triple-bubble in $\mathbb{R}^3$. Right: the $2$D cross-section through its plane of symmetry.
• Figure 2: A standard triple-bubble in $\mathbb{S}^2$; also, the cross-section of a standard triple-bubble in $\mathbb{S}^3$ through its hyperplane of symmetry.
• Figure 3: A standard quadruple-bubble in $\mathbb{R}^3$ (also, the cross-section of a standard quadruple-bubble in $\mathbb{R}^4$ through its hyperplane of symmetry) from different angles.
• Figure 4: A perpendicularly spherical Voronoi cluster $\Omega$. Top left: $\Omega^\mathbb{S}$ on $\mathbb{S}^2$ drawn with the North pole $N$ at the top (the equator of symmetry is the central parallel); Top right: $\Omega^\mathbb{R}$ on $\mathbb{R}^2$ after stereographic projection from a pole on the equator; Bottom left: $\Omega^\mathbb{S}$ drawn with North pole in the center (the equator coincides with the circumference); Bottom right: the orthogonal projection $\Omega^\mathbb{B}$ of $\Omega^\mathbb{S}$ onto the equatorial disc $\mathbb{B}^2$ consists of convex polyhedral cells (colors lightened for better contrast).
• Figure 5: The (constant) quasi-center of a spherical cap on $\mathbb{S}^n$ is $\mathbf{c} = \mathfrak{n} - \mathbf{k} p$; the usual center is at $-\mathbf{c}/\mathbf{k} = p - \mathfrak{n}/\mathbf{k}$.

Theorems & Definitions (245)

• Definition 1.1: Standard Bubble on $\mathbb{R}^n$ and $\mathbb{S}^n$
• Conjecture : Multi-Bubble Isoperimetric Conjecture on $\mathbb{R}^n$
• Conjecture : Multi-Bubble Isoperimetric Conjecture on $\mathbb{S}^n$
• Theorem : Multi-Bubble Isoperimetric Conjecture on $\mathbb{G}^n$
• Definition 1.2: $\mathbb{S}^m$-symmetry
• Theorem 1.3: Perpendicular Sphericity
• Definition 1.4: Spherical Voronoi Cluster
• Remark 1.5
• Remark 1.6
• Definition 1.7: Perpendicularly Spherical Voronoi Cluster