Stable Gaussian minimal bubbles

Abstract

Abstract It is shown that 3 disjoint sets with equal Gaussian volumes that partition $$\mathbb {R}^{n}$$ R n with nearly minimum total Gaussian surface area must be close to adjacent 120 degree sectors, when $$n\ge 2$$ n ≥ 2 . These same results hold for any number $$m\le n+1$$ m ≤ n + 1 of sets partitioning $$\mathbb {R}^{n}$$ R n , conditional on the solution of a finite-dimensional optimization problem (similar to the endpoint case of the Plurality is Stablest Problem, or the Propeller Conjecture of Khot and Naor). When $$m>3$$ m > 3 , the minimal Gaussian surface area is achieved by the cones over a regular simplex. We therefore strengthen the Milman–Neeman Gaussian multi bubble theorem to a “stability” statement. Consequently, we obtain the first known dimension-independent bounds for the Plurality is Stablest Conjecture for three candidates for a small amount of noise (and for $$m>3$$ m > 3 candidates, conditional on the solution of a finite-dimensional optimization problem). In particular, we classify all stable local minima of the Gaussian surface area of m sets. We focus exclusively on volume-preserving variations of the sets, avoiding the use of matrix-valued partial differential inequalities. Lastly, we remove the convexity assumption from our previous result on the minimum Gaussian surface area of a symmetric set of fixed Gaussian volume.

The authors' abstract, as published at the source. Calculus of Variations and Partial Differential Equations, 2026 · DOI ↗