The Gaussian Double-Bubble and Multi-Bubble Conjectures

We establish the Gaussian Multi-Bubble Conjecture: the least Gaussian-weighted perimeter way to decompose Rⁿ into q cells of prescribed (positive) Gaussian measure when 2≤ q≤ 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 Rⁿ (n≥ 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∘ 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.