The Structure of Gaussian Minimal Bubbles

Abstract

It is shown that m disjoint sets with fixed Gaussian volumes that partition $$\mathbb {R}^{n}$$ with minimum Gaussian surface area must be $$(m-1)$$ -dimensional. This follows from a second variation argument using infinitesimal translations. The special case $$m=3$$ proves the Double Bubble problem for the Gaussian measure, with an extra technical assumption. That is, when $$m=3$$ , the three minimal sets are adjacent 120 degree sectors. The technical assumption is that the triple junction points of the minimizing sets have polynomial volume growth. Assuming again the technical assumption, we prove the $$m=4$$ Triple Bubble Conjecture for the Gaussian measure. Our methods combine the Colding–Minicozzi theory of Gaussian minimal surfaces with some arguments used in the Hutchings–Morgan–Ritore-Ros proof of the Euclidean Double Bubble Conjecture.