The Gaussian Double-Bubble and Multi-Bubble Conjectures

Abstract

We establish the Gaussian Multi-Bubble Conjecture: the least Gaussian-weighted perimeter way to decompose $\mathbb{R}^n$ into $q$ cells of prescribed (positive) Gaussian measure when $2 \leq q \leq n+1$, is to use a "simplicial cluster", obtained from the Voronoi cells of $q$ equidistant points. Moreover, we prove that simplicial clusters are the unique isoperimetric minimizers (up to null-sets). In particular, the case $q=3$ confirms the Gaussian Double-Bubble Conjecture: the unique least Gaussian-weighted perimeter way to decompose $\mathbb{R}^n$ ($n \geq 2$) into three cells of prescribed (positive) Gaussian measure is to use a tripod-cluster, whose interfaces consist of three half-hyperplanes meeting along an $(n-2)$-dimensional plane at $120^{\circ}$ angles (forming a tripod or "Y" shape in the plane). The case $q=2$ recovers the classical Gaussian isoperimetric inequality. To establish the Multi-Bubble conjecture, we show that in the above range of $q$, stable regular clusters must have flat interfaces, therefore consisting of convex polyhedral cells (with at most $q-1$ facets). In the Double-Bubble case $q=3$, it is possible to avoid establishing flatness of the interfaces by invoking a certain dichotomy on the structure of stable clusters, yielding a simplified argument.