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

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 $q\le n+2$. The double-bubble conjecture on ${\mathbb{R}}^3$ was confirmed in 2000 by Hutchings-Morgan-Ritoré-Ros, and is nowadays fully resolved for all $n\ge 2$. 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 $q\le \min (5,n+1)$, namely: the double-bubble conjectures for $n\ge 2$, the triple-bubble conjectures for $n\ge 3$ and the quadruple-bubble conjectures for $n\ge 4$. In fact, we show that for all $q\le n+1$, 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 $q\le n+1$ 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.

91 pages, 14 figures. Made some final corrections