The Structure of Gaussian Minimal Bubbles

TL;DR

This paper proves that minimal Gaussian surface area partitions of space into disjoint sets are essentially flat, confirms the Gaussian Double Bubble conjecture for three sets, and extends results to four sets under certain conditions.

Contribution

It establishes the flatness of minimal Gaussian bubbles and proves the Gaussian Double Bubble and Triple Bubble conjectures for four sets, using novel second variation and geometric analysis techniques.

Findings

Minimal Gaussian bubbles are $(m-1)$-dimensional.

Confirmed the Gaussian Double Bubble conjecture for three sets.

Extended the Triple Bubble conjecture to four sets under technical assumptions.

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-Ritor\'{e}-Ros proof of the Euclidean Double Bubble Conjecture.