The Structure of Isoperimetric Bubbles on ℝ^n and 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 ≤ n+2. The double-bubble conjecture on ℝ^3 was confirmed in 2000 by Hutchings-Morgan-Ritoré-Ros, and is nowadays fully resolved for all n ≥ 2. The double-bubble conjecture on 𝕊^2 and triple-bubble conjecture on ℝ^2 have also been resolved, but all other cases are in general open. We confirm the conjecture on ℝ^n and on 𝕊^n for all q ≤min(5,n+1), namely: the double-bubble conjectures for n ≥ 2, the triple-bubble conjectures for n ≥ 3 and the quadruple-bubble conjectures for n ≥ 4. In fact, we show that for all q ≤ n+1, a minimizing cluster necessarily has spherical interfaces, and after stereographic projection to 𝕊^n, its cells are obtained as the Voronoi cells of q affine-functions, or equivalently, as the intersection with 𝕊^n of convex polyhedra in 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 ≤ 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 ℝ^n and 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.