Plateau Bubbles and the Quintuple Bubble Theorem on ${\mathbb{S}}^n$

Abstract

Sullivan's multi-bubble isoperimetric conjectures in $n$-dimensional Euclidean and spherical spaces assert 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 by Hutchings-Morgan-Ritoré-Ros (and later extended to ${\mathbb{R}}^n$). The double-bubble conjecture on ${\mathbb{S}}^n$ ($n\ge 2$) and the triple- and quadruple- bubble conjectures on ${\mathbb{R}}^n$ and ${\mathbb{S}}^n$ (for $n\ge 3$ and $n\ge 4$, respectively) were recently confirmed in our previous work, but the approach employed there does not seem to allow extending these results further. In this work, we confirm the quintuple-bubble conjecture on ${\mathbb{S}}^n$ ($n\ge 5$), and as a consequence, by approximation, also the quintuple-bubble conjecture on ${\mathbb{R}}^n$ ($n\ge 5$) but without the uniqueness assertion. Moreover, we resolve the conjectures on ${\mathbb{S}}^n$ and on ${\mathbb{R}}^n$ (without uniqueness) for all $q\le n+1$, conditioned on the assumption that the singularities which appear at the meeting locus of several bubbles obey a higher-dimensional analogue of Plateau's laws. Another scenario we can deal with is when the bubbles are full-dimensional ("in general position"), or arrange in some good lower-dimensional configurations. To this end, we develop the spectral theory of the corresponding Jacobi operator (finding analogies with the quantum-graph formalism), and a new method for deforming the bubbles into a favorable configuration. As a by-product, we show that the Jacobi operator on a minimizing configuration always has index precisely $q-1$ and hence the corresponding isoperimetric profile is concave, answering a question of Heppes. Several compelling conjectures are proposed, which would allow extending our results to all $q\le n+1$ unconditionally.

82 pages, addressed the slight difference in conventions from arXiv:2205.09102, added reference to Heppes' question on the concavity of the isoperimetric profile and additional recent references regarding isoperimetric problems on clusters