  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.
