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

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-Ritor\'e-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 M\"obius 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.