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- W3204346410 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 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." @default.
- W3204346410 created "2021-10-11" @default.
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- W3204346410 date "2021-12-01" @default.
- W3204346410 modified "2023-09-27" @default.
- W3204346410 title "The Gaussian Double-Bubble and Multi-Bubble Conjectures." @default.
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