The Gaussian Double-Bubble and Multi-Bubble Conjectures

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